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Exact constructions of square-root Helmholtz operator symbols: The focusing quadratic profile


Louis Fishman , Maarten V. de Hoop and Mattheus J.N. van Stralen 



 Naval Research Laboratory, Code 7181, Stennis Space Center, MS 39529

 Center for Wave Phenomena, Colorado School of Mines, Golden, CO 80401

Plasma Optical Fibre B.V., Zwaanstraat 1, 5651 CA Eindhoven, The Netherlands

ABSTRACT

Operator symbols play a pivotal role in both the exact, well-posed, one-way reformulation of solving the (elliptic) Helmholtz equation and the construction of the generalized Bremmer coupling series. The inverse square-root and square-root Helmholtz operator symbols are the initial quantities of interest in both formulations, in addition to providing the theoretical framework for the development and implementation of the ‘parabolic equation’ (PE) method in wave propagation modeling. Exact, standard (left) and Weyl symbol constructions are presented for both the inverse square-root and square-root Helmholtz operators in the case of the focusing quadratic profile in one transverse spatial dimension, extending (and, ultimately, unifying) the previously published corresponding results for the defocusing quadratic case [J. Math. Phys. 33 (5), 1887-1914 (1992)]. Both (i) spectral (modal) summation representations and (ii) contour-integral representations, exploiting the underlying periodicity of the associated, quantum mechanical, harmonic oscillator problem, are derived, and, ultimately, related through the propagating and nonpropagating contributions to the operator symbol. High- and low-frequency, asymptotic operator symbol expansions are given along with the exact symbol representations for the corresponding operator rational approximations which provide the basis for the practical computational realization of the PE method. Moreover, while the focusing quadratic profile is, in some respects, nonphysical, the corresponding Helmholtz operator symbols, nevertheless, establish canonical symbol features for more general profiles containing locally-quadratic wells. Key words: one-way wave equations, pseudodifferential operators, normal modes

1. Introduction The global nature of wave propagation problems, as modeled by the elliptic scalar Helmholtz equation, renders the computational solution quite difficult in extended inhomogeneous environments (1, 2). The development and application of the ‘parabolic equation’ (PE) method (3, 4, 5, 6, 7) has successfully addressed this issue for appropriately, weakly range-dependent environments, where, for the most part, one-way (forward) wave fields are computed and back-scattered energy is neglected. In recent years, the PE method has been extended to fully-coupled, two-way, elliptic wave propagation through two complementary approaches: (i) the exact, well-posed, one-way reformulation of elliptic wave propagation problems (1, 2, 8) and (ii) the construction and application of the generalized

Bremmer coupling series (9, 10, 11). Both methods of extension are based on ideas and constructions from wave field decomposition, invariant imbedding (reflection and transmission operators), and the closely related Dirichlet-to-Neumann (DtN) operators, and make use of micro-local analysis (pseudodifferential and Fourier integral operators and path integrals). While the current focus in the direct and inverse analysis of the above two approaches is primarily on the properties and singularity structure of the scattering (reflection and transmission) and DtN operator symbols (1, 12, 13), the inverse square-root and square-root Helmholtz operator symbols are the initial quantities of

152

L. Fishman, M.V. de Hoop & M.J.N. van Stralen

interest in both formulations. In many respects, however, despite the detailed treatment of the fully-coupled, two-way, elliptic formulations in the above-referenced literature, the original PE method provides the most straightforward introduction to the Helmholtz operator symbols. Starting from the two-dimensional, space-frequency domain, scalar Helmholtz equation, the formally exact wave equation for propagation in a transversely inhomogeneous half-space supplemented with appropriate right-traveling-wave radiation and initialvalue conditions is given by (1, 2)



 

i  

(1)

 



is the one-way wave function,  is a reference or average wave number (proportional to frequency), and   "! # $%&(' )+* "! , .- /

#*0 ,$%* 1"243 *   (2) ) "!  is the refractive index field. The range coordinate, 5 , is associated with the is the square-root Helmholtz operator, where ! where

‘one-way’ direction; is the transverse coordinate. In the subsequent formal analysis, the transverse coordinates will also be freely denoted by 6 . !  !%7  For a point source of the volume injection type, located at 5 and , the appropriate initial-value condition or (source) decomposition is given by (1)

8    ! 9  ; 2



= "!8?!%7 @

i : 

(3)



where denotes the inverse or parametrix of (i.e., the inverse square-root Helmholtz operator). The Helmholtz operator symbols are defined in a pseudodifferential operator (operator-ordering) calculus (14, 15, 16, 17). Let A  A "!  !CB   denote the kernel associated with operator , i.e.,

 D8E   ! 9GFIH JLK ! B A "!  ! B  8   ! B M 5 5

(4)

A ; 2  A ; 2 "!  ! B 

and denote the kernel associated with the inverse operator, is defined as

7OQP R

N

 ; 2

. Then the symbol, for the inverse square-root Helmholtz

TS  9  GF H J K/UDVXW/Y  S UI A ; 2   ZUI 6 i 6 6

(5)

in the standard (left) pseudodifferential operator calculus, and as

O P R

[

TS  9  F H J K/UDV0WY  S UI A ; 2]\ ? i 6 6 *

2 U^ 6

 *

2 UI_ N

7O [

(6)

O

in the Weyl pseudodifferential operator calculus (14, 15, 16, 17). Likewise, definitions of and hold for the square-root Helmholtz operator itself. In terms of the square-root Helmholtz operator symbols, the one-way wave equation (1) can be written as `



 aFbH J 

.K S N i   :dc

7O

TS  ! eV0WY  b S ! gf    S 9 5 i

or



  aF H J K ! B F H J 

i  

 :dc

.K S [

O

(7)

 k \ S  2 "!  ! B # _hV0WYi' Sh"!8?j! B  1    ! B 9 i 5 *

in the standard and Weyl calculi (14, 15, 16, 17), respectively. Here,    !  5 with respect to the transverse coordinate,

(8)

f    S  5 is the Fourier transform of the one-way wave function

f    S 9GF H J K ! B V0WY .? Sl! B e   ! B @m 5 i 5

(9)



o

For a fixed range point, 5 5bn say, Eqs.(7) and (8) provide the basis for the nonreflecting boundary conditions ubiquitous in numerical wave field computations (1, 4, 10).

In Van Stralen et al. (10), interchange p and q , and replace

s

r (the vertical slowness operator) by t , u r by v

7O , and

w

w r 2 by x  (and r * by x ; ).

Exact square-root operator symbols

153



The fundamental solution (propagator), y , associated with Eq.(1) can be expressed as a lattice multivariate integral, with Hamiltonian equal to the square-root Helmholtz operator symbol (1,ˆŠŒ 2, 11, 18, 19, Œ 20, 21),

  ; 2 ! z B! Bg   /  {  ? B |~}€ FbH J ….† P R ‡ˆŠ‰ 

.K S^‹ ‡ ‰ K !/‹  y 5 5 5 5 k‚„ƒ  C:%c  2 ‘’Œ ‘’Œ ‘ Œ ‘’2 Œ ‘’Œ  7O Ž V0WYZ i   ‘’‰ S ‹ "! ‹ ?
)+* "! 9)+* ?+¦ * ! *› ˜ (14) )  ¦¨§ì'  £> ? M ž£ ï  1  æ i c ¿ ˜

7O

N

TS  9   6



TS   6





(33)

which is the counterpart of Eq.(28), where




? *2   

®?   V

 £>²

M £žï æ

 #£ i :  £>##* Ž : ; 2 ã 42 3 *,V0WY .? >£  «C‰ : ¬E­ ˜

²

V0WY

 V  : £# ?¹-   V  : £># 243 * ?

V  : £#    : «£> À ?   : £> : i « i

' ?„ ¬  -]?M 6 :

(35)

and the principal value of the square-root function appearing in Eq.(33) is taken. 7O N TS   6 derived by Van The result in Eq.(33), in essence, supplements the approximate, truncated series representation for Stralen (4), with an integral which results in the exact expression. As such, the expression in Eq.(33) is well-defined in the limit š æ , the nonconvergent behavior of the (first) series being exactly cancelled by the appropriate contribution from the integral £   term in the neighborhood of . The details of this cancellation will result in computationally convenient operator symbol £g '  š  representations, thus, focusing on the neighborhood of , partitioning the integral on in Eq.(33) into the contiguous sets '  1 ' M š   and with ç , and noting that

 



 ˜



°3*

N

allow for

N

7O



K£e£0; 243 *bV0WY ë® £# M £žï

F :



TS

9

« À

(36)

«

ã ; 2 - ' ?„ TS   -]?®  1 ¾ i 243 *  «C‰ : ¬ 6 ¬E­ ^ : ¿ ˜ 7O TS   6 to be written in the form ; 2  9   243 * «¥‰ ' ?„ ¬  -Q?à®  1"243 *a¾ 6 :d¶ : ¬E­ ˜ 243 * ' ?„  -Þ?à®  1"243 * Ž -  : i¾ : ¬ « À c ¿ « ; 2   243 * ã «C ‰  ¬  -]?®  243 *¹¾ i i :%¶ ¬E­ : :^¿ 243 *   -Q?®  243 * í Ž -Q? : ¾ 2 2 : ¬ c ¿





$#

!



— 2

í 2

.-

«ÞÀ

"!

$#



æ

C:

TS   6

i

í

 ¬  -Þ?ñ ò ù ´Î Ø’Ù Ð Ñ The estimate,

++

5

+



!

7

98

6

S  6

  6 ó

ó

;:=


K£ £X; 243 *IV0WY ë® £#ì'  £> ? M £’ï



. º



* ¾ :

243 * V0WY,' ? ¿

 

æ

1

++

 -]? æ i ¿ c ˜

O [

O 

I

J I 

L 

(61)

where

« ã ; 2  LVXW/Y .? ¬ £> V0WY .? £> «C‰ ' ?„ ¬  -Þ?

J I 

C

 ¬  -Þ? ? M £’ï  ¸

L



&

 ¬  -]?® # :

m

The principal results of Section 2 are the Helmholtz operator symbol representations given in Eqs.(21), (47), (48), (56), (63) and (64).

3. Contour-integral operator symbol representations It follows from the construction in Eq.(57) that the inverse square-root Helmholtz operator symbol can be written in the form

STVOQUW P R TS [

 9   6

?

i

¾

-

ci¶,¿

ƒ

243 * F ˜

K£ £0; 243 *bV0WYL' ® £ ?

 



 £> 1 V

 £#M

®»º - 

(65)

162

L. Fishman, M.V. de Hoop & M.J.N. van Stralen

where there are no propagating modes. For the corresponding defocusing quadratic profile (compare Eq.(14)) defined by

)+* " ! 9  )+*  ¦ * ! *› ˜ ) ˜  ¦é§·© ª  , the operator symbol integral representation (23) with ƒ - 243 * OQP R TS  9 ? V0WY \C±° _Z¾ [ K£e£ ; 243 * V0WYL' ë® £i F 6 i c i c,¶I¿ ˜

YXZ T

\[  

where

[

¶



 £## 1 V

 £>M

(66)

; 2 ´¦ * * ?·S *’ 6

(67)

(compare Eq.(31)), in conjunction with Eq.(65), lead to the relation

STVOQUW P R TS [

 9   6

|~}~ ‚

] ]

[

XO›Z P T R TS

®

 M   6

i

ºG- 

(68)

adopting an obvious superscript notation in this section for clarity. The analytic continuation result in Eq.(68) can be extended to the square-root Helmholtz operator symbol following the construction in Eq.(61) and noting the corresponding integral representation for the defocusing quadratic profile given in Fishman (23). The principal goal in this section is to extend the integral representation in Eq.(65) and the subsequent analytic continuation result in Eq.(68) connecting the focusing and defocusing quadratic profiles (and the analogous results at the level of the square-root ®»º® § '   š  Helmholtz operator symbol) from to , thereby explicitly accounting for the presence of the propagating modes. ¦ The basic idea is to modify an -rotation/contour-integration construction, which underlies the Eq.(68) analytic continuation result ® º for , by explicitly incorporating the periodicity of the associated, parabolic (Schr¨odinger) equation fundamental solution (propagator) into the Helmholtz operator symbol construction procedure inherent in Eq.(66).

¦

A. The complex -rotation and contour integration: analytic continuation for

¦

®åº

®

º -

-

First, the -rotation/contour-integration construction for the case will be outlined, followed by the explicit incorporation of the underlying periodicity to produce the modified construction leading to the desired integral representations for the Helmholtz ® § '  š  operator symbols for . The extended analytic continuation results, following directly from the integral representations, will then be established. ¦ For the -rotation/contour-integration construction procedure, the starting point is the exact, closed-form expression for the Weyl symbol for the inverse square-root Helmholtz operator in Eq.(66), with the aid of Eqs.(18)-(19), written as (note the change £ £ ¦ of integration variable,  )



243 *

ƒ


i   )+˜ * £i\[    ´¦ £## V ´¦ £>Xm (69) ¦ In Eq.(69), let   ¦ ¦ *  *  ?·S *¦ 1 V0WY  ib  ,  - b º ch[: ,  with the corresponding transformation [  [   I' VXW/Y  ; 2 V0WY .? ¶ ib : ib 6 , so that in the limit b
L c c^C: ,  i . Consider the contour integral g *  f h 4 2 3

[ed ? V0WY \ °± _e^  KiYii; 243 *bV0WY
 )+* ¼ i D[   ´¦  li # V ´¦  bi M c i i  ˜ (70) ca` ± 6k 'k 'k 'k i * ° 2 where j as illustrated in Fig. 1. For the integrand in Eq.(70), in the complex -plane, the branch point i ; 243 * (associated with ) is at the origin with the branch line chosen to lie along the negative real-axis, and the isolated singularities l ) are located at the points (associated with the zeros of i  ¦ V0WYKJ i ¾ c ? b L  : ¬  -  c  ¬  ™’™’™  ? :  ?L- Ÿ -  :  ™0™ž™ m  (71) : ¿ : ¦ k k The -rotation and subsequent contour integration have been specifically constructed so that the contributions from 2 and ° [

XZ T

OQP R

TS  9   6

? V0WY \C°±

c i

_^ _





a` c

F

K£e£0; 243 *IV0WY

˜

will ultimately result in the operator symbol for the focusing case and the analytic continuation of the operator symbol for the defocusing case, respectively. Application of the Cauchy integral theorem (35) followed by standard arguments (35) to establish ± ª š º ® ºG  that the contributions from * and vanish, respectively, in the and limits for ch: (Fig. 1) and result in the equality

k

? VXW/Y \ °±

c i

F^ _

c



`

243 * F

ƒ

k

˜




i  / )

K£e£ ; 243 * V0WY

˜

[   

* £i

m_

 V

´¦  £##

´¦  £>9

- b

(72)

Exact square-root operator symbols

163

Imτ τ - plane

π −ϕ 2

branch point branch line Γ4

ρ

Γ1 −ϕ

R

Reτ

Γ2

Γ3 isolated singularities Ö

n

s

s

s ±

s

2 Ð * Ð ° Ð Figure 1. The contour of integration and the integrand singularity structure in the complex -plane for the Helmholtz operator symbol construction in Eq.(70).

o

? VXW/Y \ °±

c i

ƒ 243 * VXW/Y ? K £e£0; 42 3 , * VXW/YL'  VXW/Y .?  ® £i \ 2 _&F i i * i ci¶I¿ ˜ º 2  » ® º - m  * c -

_ supplemented with an outgoing-wave radiation condition (1, 23). The parabolic equation fundamental solution,   !/z B  ! B  5 related to 5 through (23)

q

ƒ * ;b° 3 *8¾ i 243 F K£e£X; 243 *IV0WY :

 c ¿ ˜

p *2 i 

  !/z  ! B g 5

and satisfies



ut

 I .-

i 

*  $ *  C:  *

 s

2  ) * "!  ?¹- 

 £i

 £X !/z  ! B 9 

5

 s

*Ÿ£0; 2 

 £0 !/z   ! B M

s

 £0 !/z   ! B 

(74) say, is

(75)

(76)

supplemented by

s

  !/z  ! B g = "!?
(77)

164

L. Fishman, M.V. de Hoop & M.J.N. van Stralen

For the focusing quadratic profile, the parabolic equation fundamental solution takes the form implied in Fishman (23, (A1)) and given in Schulman (36, p.38),

v^

 £0 !z  ! B 9

s

A

Ž V0WY

¦

 } ´¦ £>



:%c i

#  )+˜ *

2

* i

?¹-  £i

243 *

`

¦

}

´¦ £>

 ' ! *& "! B #* 1

l %´¦

£# ? :

&RC

!¥! B 



(78)

in accordance with the corresponding quantum mechanical harmonic oscillator formulation (27, 36). This function is, to within an  ¦  £ . Substituting Eq.(78) into Eq.(75), the subsequent result into Eq.(73), exponential phase factor, periodic in with period :%ch  c^C: : and applying the transform (6) lead to the lhs of Eq.(72) ‘evaluated’ at

STVUW

xw

OQP R

[

TS   6

where

? V0WY \C°±

y

? )+*  i ˜



c i

? ¦&® i

e^



_

243 *



F

a` c

˜

z

and

ƒ

b

ry £>uz

K£e£X; 243 *IVXW/Y .?

´¦ £>EÁV0WYL' ?

i

´¦ £#&

(79)



´¦ >£  1 V ´¦ £#@m

{

The key to the extension of Eqs.(65) and (68) is the reduction of this semi-infinite integral to an integral over a single period through the application of the following theorem. Theorem. Let

ƒ

F ˜

z

´¦ £>



be a :%ch

ry £#z

K£e£0; 243 *bV0WY .?

¦ 

£ §|z ´¦ *~} 3 ] K£ µ/.-

K £e£0; 243 *bV0WY .?

˜

¦

243 * F

£>9

y

ry |z

¦ #&VXW/Y .? £>  ´¦ £ X V0WY .? C: C:%c :%c  :%c ¿ ˜ µ  •   « œ is the Lerch transcendental where function defined by (29, 37) ƒ µ   •  9 «¥‰  ¬  •  ;  •   ?L-  ?  ™’™’™ ï ï º - m œ œ : œ ˜ ¾

´¦ £>M

(81)





€

G

(82)

The Lerch transcendental function can be analytically continued into the cut complex œ -plane via the integral representation (Erd´elyi et al. (37, p.27))



µ   •  g   œ

-

k   

ƒ

F ˜

~€

V0WY . ? • #£   -Þ? 0V WY .? £> œ š

K£e£ ; 2

œ not on the real axis between and .™ 

ª V • “

,



k ª V

–¼ç

“

–¼ç

k

k is the gamma function (29, 37). It is continued • and where in via the relationship given in Erd´elyi et al. (37, p.27), « ; 2 µ   •  9 µ  „ƒ  •  i ‚ «C‰  ¬  • X;€ …ƒ  -   ™’™ž™ m œ œ2‚ œ œ : ˜

(83)

(84)

(A more detailed treatment of the Lerch transcendental function can be found in the references (29, 37).) Proof. Starting with the integral on the rhs in Eq.(81) and applying Eq.(84) yield

¾

¦

:dc›¿ ¾

243 * F ¦ :%c¿

~} ] * 3

˜ 243 * F

µ Œ .-  ´¦ £ K£ X.VXW/Y .? C : C:dc :dc

~} ]

K£ µ/.-

jF

‚ ~* } 3 ]

‹  2 * 3

‚

‚ ~* } 3 ]

˜

C:

 ´¦ £

C:%c

y

ry £#z

¦ #&V0WY .? 

X V0WY .?

:%c

ry £>uz

K £e£0; 243 *bV0WY .?

y 

´¦ £#g

ry £>|z

¦ #&VXW/Y .?

…ƒ

´¦ £>&

ê - 

´¦ £>

(85)

Exact square-root operator symbols

165

Im τ τ - plane

π −ϕ 2

branch points ρ

Γ1’

Γ4’ branch lines

−ϕ

2π/ω

Re τ

Γ2’

Γ3’

isolated singularities Ö

n

B

s B

s B

s B

s B±

Figure 2. The contour of integration and the integrand singularity structure in the complex 2 Ð * Ð ° Ð -plane for the Helmholtz operator symbol construction in Eq.(88).

o

z

exploiting the periodicity of . The second integral on the rhs of Eq.(85) can be written as

‚ ~* } 3 ]

F ˜

ƒ F ˜

ry £>|z

K £e£0; 243 *bV0WY .?

ry £#|z

´¦ £>9

K£e£X; 243 *IV0WY .?

ƒ

´¦ £# ? F

‚ ~* } 3 ]

ry £>uz

K £e£0; 243 *bV0WY .?

´¦ £>Mm

(86)

z

On the other hand, consider Πthe first integral on the rhs of Eq.(85). Substituting Eq.(82) into this integral, interchanging the order of integration and summation, and exploiting the periodicity of result in

¾

~} ]

‹  2 * 3 ¦ # &V0WY . ? £# ´ ¦ £# K £ µ .-  ´¦ £  X.VXW/Y . ? « Œ C: C:dc d: c  :dc›¿ * 3 ƒ ƒ ‹  «¥Œ  2 * 3  C « ‰  F K£e£X; 243 *IVXW/Y .? £> ´ ¦ >£ 9GF K £e£X; 243 *I0V WY .?  ‹  * * 3 3 ˜ ¦

243 * F

‚

y

‚ ~} ] ~} ] ‚ ‚ ~} ]

ry uz

Combining Eqs.(87), (86) and (85) results in the lhs of Eq.(81).

ry uz

‚ ~} ]

ry £#|z

´¦ £#Mm

(87)

C. Symbol contour-integral representations for the focusing case While the semi-infinite integral in the operator symbol construction for the focusing quadratic profile is not well defined at present, ¦ ® ºthe above theorem and the -rotation/contour-integration construction for the case motivate the following conO›P previous R TS [   6 . Consider the contour integral (compare Eq.(70)) struction procedure for

TVUW

[ B

d

? V0WY \C°±

fh‡ Ki

c

†^ i _



a` c

243 *



¾

¦

243 *

d: c›¿ X V0WY 


i   ) ˜ * i¼\[    ´¦  ib# V ´¦  il (88) ® G G   -  :  ™’™’™ , where the contour j B k B 2 \k B* †k B° \k B± is defined in Fig. 2. For the integrand in Eq.(88), in the complex for µ/.-  ´¦ i ® # i -plane, the branch X V0WY  points associated with are located at the points C: C:%c :%c i i ? :d¦ c ¬  ¬  -  :  ™’™ž™  (89) Ž¹Ž

µ/.-

C:

 ´¦

i

C:%c

:%c i

® #eV0WY

166

L. Fishman, M.V. de Hoop & M.J.N. van Stralen Im τ τ - plane 2π ’ Re τ

2π

Figure 3. The contour of integration Eq.(91).

ˆ B

o

in the complex -plane for the Helmholtz operator symbol representation in

i

i

with the associated branch lines chosen to lie along the negative real -axis, and the isolated singularities are located as in the ® ºGµ .-  ´¦ ® # X.V0WY  . The series representation in Eq.(82) for is uniformly and previous construction for the case C: :dc :dc i ®  ï ï º ï ïC   ÁV0WY  absolutely convergent for œ , conditionally convergent for œ and œ , and divergent for œ , where œ :dc i . Thus, for the focusing quadratic profile, the Lerch transcendental function in Eq.(88) is defined by the series in Eq.(82), residing on ®  -   ™ž™’™  : the circle of convergence, away from œ , i.e., . B± Application of the Cauchy integral theorem (35) to Eq.(88) followed by an evaluation of the contribution along in the  limit result in

G

G

m‰

e^ ¦  :` *~} 3 ] K £ µ .-

? V0WY \ ±° Ž F 

c i

˜

? V0WY \C°±

~} *

Ž F ˜

k

243 * -

_

c : -

 ´¦ £

:dc

X.V0WY 

:%c i


i   )

® #eV0WY

˜

[    

* £i

 V

´¦  £>#

V0WY .?   i :d¶ 42 3 * c ® #VXW/YL'  VXW/Y .? K£ µ .-   £  V0WY .? X#VXW/Y   ® £i C: C:%c :dc i i i i

´¦  £#

_ c i 

b

b

‹Š Œ Œ Œ \[   

TVU%W

Œ



D[   

b

 £## 1 V

 £>

® # _ ¾ : 243 * F K V0WY .? ¥ µ/.- ŸV0WY .? CX#V0WY  c i C: :%c i i i ¶ ¿ ˜   V0WY^'  ? ¥ 1 # 1 V  V0WY,'  ? ¥ 1 M Ž V0WYL' i  d: c ® V0WY .? i¥, :dc :dc i i (90) º ®    -   ™ž™’™  ch: , : . While theO›lhs Eq.(90) is not well defined, at present, in the limit ch: , the rhs in Eq.(90) is P R Tin S   [   c^C: , suggesting that 6 is given by the rhs of Eq.(90) evaluated at ch: , i.e., continuous at OQP R TS  g ? V0WY \ ± 2 _ [ 6 c i   243 * (91) :d¶ c +V0WY \ ± 2

- b

G

b

STVU%W

Ž ‡ K i

b

Œ



Œ

b

b

bBc

i ?     ib 1 V  il@ ®  G  -  :  ™ž™’™  ª V i B i i  to  i  :%c and the circular arc, ï ihïb :dc , where the contour  in the complex -plane consists of the – -axis from D i i ß  from :%c to i :%c , as illustrated in Fig. 3. Application of “ the Cauchy integral theorem (35) to Eq.(91) then yields the final Ž F

µ/.-

C:

 .?

iC:dc

i

 /.V0WY 

:dc i

® #eV0WY¼' ®

integral representation

[

VTOQU%W P R TS Ž F

 g   6

 uK i

 42 3 * d: ¶ c µ .-  .? ® # V0WYL' ®  />V0WY  : :dc i i :dc ? V0WY \ ± 2

_ c i 

i

(92)

i ? {    li  1 V  bi & ®

G

  -   ™’™’™  :

Exact square-root operator symbols

167

Im τ τ - plane

3π

2π

π

Re τ

branch point isolated singularities branch line

ˆ

−π

Figure 4. The contour of integration and the integrand singularity structure in the complex Helmholtz operator symbol representation in Eq.(92).



where the contour in Eq.(92) starts at points at (cf. Eq.(89))

i

?

iZ 

and ends at

iZ

o -plane for the

i :%c , keeping the integrand singularities, which include branch

¬  -  :  ™0™’™ 

 i :%c ¬

(93)

i with the associated branch lines chosen to lie on the negative imaginary -axis, and isolated singularities at (cf. Eq.(71)) i i  : ¬  -  c  ¬  ™’™ž™  ? :  ?L- > -  :  ™’™0™  (94) : ª V i  í ‘outside’ the contour with respect to the half-plane –ìç , as illustrated in Fig. 4. In Eq.(92), the occurrence of I defined in Eq.(49) is recognized. Now, the specific choice, ‘“’ , for“ the contour  , shown in Fig. 5, provides the appropriate definition for the lhs integral in Eq.(90) in the limit b
L c ch: , viz., * 4 2 3 ¦

? V0WY \ ±° _e^  c i :` c *~} 3 ] ® #eV0WY
 )+* £i[  ´¦  £># V ´¦  £# K £ µ .-  ´¦ £ X.V0WY  Ž F : :dc :%c i i  ˜     



˜  ? V0WY \°± c i_  :d¶  243 * c *~} ® #eV0WYL' ë® £ ? K£ µ .-   £ X.V0WY  Ž F i C: C:%c :dc i {   £># 1 V  £>M ˜

where the path of integration along (cf. Eq.(79)).

'  

:dc

1

is understood to pass ‘below’ the integrand singularities at

i

£a

c^C: and ch:

ƒ

Verification of Eq.(92) – The operator symbol representation given in Eq.(92) can be verified in the following manner. Choosing ª V   and avoid the integrand singularities, and applying Eq.(84) for to the contour to lie entirely in the half-plane –¼ç “ Eq.(92) result in



? V0WY \ ± 2

_

c i 

-

 :%¶ 42 3 * c

F

 uK i

µ .-

:

 .?

i C:%c

i

 #V0WY 

:%c i

® #eV0WYL' ®

i ?     li  1 V  bi 

168

L. Fishman, M.V. de Hoop & M.J.N. van Stralen Im τ 2π

τ - plane

3π/2+δ 3π/2 3π/2−δ

τ’

π π/2+δ π/2

τ

π/2−δ Re τ

branch point branch line −π/2

™•=—›š

Figure 5. The Û contours of integration

isolated singularities

ۖ•=—a•K˜ o ” ’ and ” ’ in the complex -plane used for the proper definition of the left-Ó’Ø B

hand side integral in Eq.(90) and the high-frequency asymptotic constructions in Eqs.(129) and (147); Ñ for the specific asymptotic evaluations. in general and

? 

i

¾

-

 i›i

243 * F LK

i ?     bi  1 V  li  ? V0WY \ ±2 c i_ V0W Y  :dc i* ®  :d¶ 243 c ® # V0WYL' ® i ? i>V0WY  G   -  :  ™’™’™ :%c i {    il 1 V  ibM ® 

; 243 * V0WY¼' ®

c,¶I¿ Ž F K µ .- C:  -Q?¹ i C:%c

 i

m

(95)

Examining the first term on the rhs in Eq.(95) and applying the generating function result in Eqs.(51) and (52), interchanging the order of integration and summation, which is justified by the uniform convergence of power series within their radius of convergence and the application of the Riesz/Young theorem (31, 32), and carrying out the remaining -integration result in

i

?

i

-

¾

c,¶ ¿

243 * F

 K iYi,;

243 *IVXW/YL' ® «



* ƒ ? VXW/Y \°± _ : 3 «¥ ‰ c i 243 * ¶ ˜

œ

ŽO

i ?    bi  1 V  ib   í .- z zd?  2 2 C: C: i :dc

 ¬  -Q?à® #M :

(96)

utilizing the complex form of the integral representation of the incomplete gamma function expressed as a confluent hypergeometric function (29) given in Eq.(38). Applying standard confluent hypergeometric function identities (29, 37) then yields the expression

«   _ : - ?®  1 42 3 * '  ¬  ] c i 243 * ¶ ˜ i : ®  «Cƒ ‰ VXW/Y  ßV0WY \°± _ :dc i   243 * '  ¬  -]?®  1 42 3 * c i : ci¶ ˜ i : .™~z’™~zž™  ƒ «¥‰

? V0WY \ ±°



O 

(97)

«

  O  

.-

C:

zž-

z   -]?V0WY .?

:%c

\[  

® #eV0WY„' ë® £i i

c i

_„¾

[ 





 £## 1 V

243 * F ƒ  £># 1 V K £e£0; 243 , * VXW/YL' ë® £i i c,¶ ¿ ˜ ƒ ® #eV0WYL' ë® , K£ µ .-  -Þ?¹ 4£X#V0WY . ? £  C : i C:%c :%c i

˜

š



K£ µ/.-





\[  

G

®

km

i

ƒ

in the complex -plane can be specifiš  . Applying Eq.(84) for to

 £#

®  V0WY . ? :dc   42 3 * :%¶ c  £## 1 V  £>M ®   k  £> ?

i



G

(106)

XZ T

is taken. Following from the exact operator symbol representations for O the case established by when the limit P R defocusing TS   [ Fishman (23) and given in Eq.(66), the first term on the rhs in Eq.(106) is seen to be exactly 6 . It is readily established BB ª š that the contribution to Eq.(105) along * vanishes in the limit . Utilizing the periodicity of the hyperbolic functions,

 

"! 

i :dc

9

k



"! 

and

V

"! 

i :dc

V

9

k in the evaluation of the contribution to Eq.(105) along



"! M

BB ° then results in the term

ƒ ®  V0WY . ? ® #V0WYL' ë® £i F K£ µ/.-  -]?é 4£0>V0WY .? %: c C: :dc i   243 * i :dc i :d¶ c ˜ ª š

[ 



(107)



 £># 1 V

 £>M ®

G

k

(108)

in the limit . Adding the terms in Eqs.(106) and (108), in view of the integral representation in Eq.(66), then establishes BB  the inverse square-root Helmholtz operator symbol representation given in Eq.(105). (Note that specifying the contour in ®»º O recovers the contour-integral representation in Eq.(65) for Eq.(92) for the focusing case .) TS   [ The construction of 6 via the Weyl composition equation (23), in a manner analogous to Eqs.(101)-(102), then provides the representation

[

XZ T

XO Z T TS

 9   6

?

243 * -

¾ ¶

:^¿

c

F

 K i

µ/.-

C:

 .?

iC:dc

i

 />VXW/Y .?

:dc

® #



j

i [     li # 1 V  li Y i®

Ž V0WYL' i ë® ¼

Exact square-root operator symbols



i

[  V  bi ##* ?     bi 9

G

®

km

171 (109)

Standard symbols – The corresponding integral representations for the standard operator symbols again follow from Eq.(12) in the form

%Ÿ XZ T

7 N O P R TS  9   6




Ž V0WY and

N

%Ÿ XZ T TS 7O

 g   6 Ž V0WY Ž ® i

 i

i

® #  F K µ .-  . ?  /#V0WY .?  42 3 * i :dc C: :dc d: ¶ c ë ® ¼ 2  bI  V  l ?¹- #  V  l# 243 *+ : : : i *

?

i

i

  i

[  i ; K i

   i

® # i :dc i *2 [   : ibI  V  : li  ?¹- #   V  : il# 243 * i

V  : il   : ib ?    : il    [  V  ib# * ? i : i

: 243 * -

? ¾ ¶

:i¿ ë® ¼

E. Analytic continuation for

c

µ .-

F

C:

 .?

iC:dc

G

®



(110)

 />VXW/Y .?

®

G

km

(111)

®»§ '  š 

The contour-integral representations for the Helmholtz operator symbols given in Eq.(105) and Eqs.(109)-(111) for the defocusing quadratic profile and Eq.(92) and Eqs.(102)-(104) for the focusing case, in conjunction with the appropriate analytic structure in ¦ the complex variable associated with the contour integrals, immediately establish the extended analytic continuation results

STVOQUW P R TS O [ST U%W TS [

%Ÿ TVU%W %Ÿ T U%W

while

 ²   6  ² 6

7O P R T S  ² N Q 6 7O TS  ² N   6

[

|€}~

]‚ ] |€}~ ‚

i

] ]

[

XOQZ P T R TS XO Z T TS

i

Ÿ XZ T ] ] Ÿ XZ T ] ] |~}~ ‚ |~}~ ‚

i i

®Â§ '   š M ®

 M   6  M 6

7O P R TS  M N › 6 7O TS  M N   6



G

4

-    ™’™ž™ 

®Â§ '   š M

®

®»§ '  š M ®

(112)



(113)

G

4

-    ™’™’™ 

§ '  š Mm

(114) (115)

The equalities in Eqs.(112)-(115) extend the applicability of the previous analytic continuation results summarized in Eq.(68) ®åº ® § '  š  ® from to , with the extension to the previously omitted, integer values following from the spectral (modal) summation representations and several additional contour-integral representations briefly discussed in Appendix B. The principal results of Section 3 are the Helmholtz operator symbol representations in Eq.(92) and Eqs.(102)-(104) for the focusing case, Eq.(105) and Eqs.(109)-(111) for the defocusing case, and the analytic continuation formulas in Eqs.(112)-(115) connecting them.

4. Time-Fourier versus time-Laplace domain In the analysis of two-way wave scattering, the generalized Bremmer series that couples the one-way waves plays a fundamental role. The convergence properties of this series are understood in the time-Laplace domain (De Hoop (11)) and require Sobolev order estimates of the square-root ‘Helmholtz’ operator uniform in the Laplace parameter. However, most algorithms that compute terms in the generalized Bremmer series are carried out in the time-Fourier domain. The quadratic profile provides a canonical medium in which the transformation of the square-root ‘Helmholtz’ operator from the time-Fourier to the time-Laplace domain, and vice versa, can be carried out explicitly and understood. In the space-Laplace domain, the formally exact wave equation for (one-way) propagation in a transversely inhomogeneous half-space supplemented with appropriate right-traveling-wave radiation and initial-value conditions is given by (11)

.- l

e /8· 7 # 1 V  £> c ¿ ˜  m Ž ®    V  £>##*9   £>< (compare Eq.(125)). For the focusing quadratic profile, it likewise follows that OS¢ OS¢ O O [STVUW TS  9 |~}~ [ XZ T TS  g ? [STVUW TS  & ? [ XZ T TS  Xm |~}~ |~}~ 6 6 6 6 i i ]‚ ] ] ‚ ];¤ ¥¡¦ ‚ ¥¡¦ ] ‚ ; ];¤ ¥¡¦ ‚ ¥¡¦ [

 g  ÁV0WY \ ? ± 2 _j¾ ¶ 6 c i

243 * -

:,¿

i

c

i

F

i

(120)

(121)

(122)

i

Substituting the representations (120) and (121) into Eq.(122), then results in the equivalent expressions

STVOSU%W¢ TS [

STVOSU%W¢ TS

and

[

 g  ¾ ¶ 6

243 * -

:,¿ c Ž VXW/YL' i ë®

 Ki µ/.- C:  .? iC:%c i>V0WY .? i ? [    ib# 1 V  bi  ? i® F



ƒ  g  ÁV0WY \ ± 2 _Z¾ ¶ 42 3 * F K £e£0; 42 3 *bV0WYL' ë® £ ? 6 c i i c›¿ ˜  V  £># *   £# jm Ž ? ®  i

i

[  

  

® #

:%c i

[  V  li ##*E   li   [  



 £># 1 V

 ®

G

k

(123)

 £> (124)

The operator symbol representations in Eqs.(120) and (123) will prove useful in the construction and analysis of the square-root ‘Helholtz’ operator in the right-half of the complex Laplace plane, which will be presented elsewhere.

5. Asymptotic operator symbol expansions The contour-integral representations for the Helmholtz operator symbols developed in Section 3 enable the high- andO low-frequency, TS   [ 6 will be asymptotic operator symbol expansions to be derived in a straightforward fashion. Only the expressions for presented; the other cases can be derived in a similar manner. (The superscript notation introduced in Section 3 will now be suppressed since all subsequent results will apply to the focusing case.)

TVU%W

Exact square-root operator symbols

173

A. Low-frequency asymptotic operator symbol expansion

š

‰

®



®



®º»-

¨

In the low-frequency limit, ¶ and , enabling to be restricted to , corresponding to ¸ , where there are no propagating modes. In the absence of propagating modes, the expression for the square-root Helmholtz operator symbol in the Weyl calculus reduces to (cf. Eq.(61) with Eq.(60)),

O

TS  9   6

[

® Ž

ƒ ¾ ¶ 42 3 * F K£ £X; 42 3 *IV0WYL' ® £ ? i c ¿ ˜  V  £###* ?  #£  j » ® º - m ?

  

?



 £# 1 V



  

 £# (125)

Expanding the exponential in the integrand and ordering the resulting terms in powers of ¶ yield a sum of integrals, each of which µ  µ  can be expressed in terms of the generalized zeta function . On the one hand, is defined as a special case of the Lerch « introduced in Eq.(82), viz., transcendental function



µ    • 9

ƒ «¥‰

€

.?   ¬  • X; ·

G

 ?L-  ? •

˜

:

 ™’™0™ 

(126)

and its appropriate analytic continuations (29, 37). On the other hand,

ƒ

F



7

 £#

˜

K£e£ ; 2 V

 £>9



:

*X;

7

k

 ¢% µ   ¢ ?¹-  -

ª V ¢

]

C:

“

µ 

is represented through the integral

k

–¼ç

(127)

following from the result in Magnus et al. (29, p.34). Employing this integral representation, and integrating by parts the various terms in the above mentioned expansion then yield

[

O

¡¨ w

A

T S   ‚„ƒ 243 * i: 6 ? ?

; 2 µ  . -  C : ±  ; µ   : C: :

©

§

µ  .?L:

C:

 -



C:

a

¶

243 *

.)+*  * µ  .?  -  ´¦ * ˜± : C : C : - .) ? 1  '  = VXW/Y   1 c i c^C: i with along the ‘semicircular segments’, then results in the explicit representation

i

[

O

ª

# } 3 *X;« K £|¬

TS  g   F 6 i

ª

 £>, F

i

„} « K£|¬ } 3 *#;« ° 3 *X;

~} *

 £>, F

|¬  £>&

K£ ° 3 *# àVXW/Y \ ± 2 _„V0WY \ 2 ® _¼¾ ¶ 243 * K V0WY = F c i * c i : ¿ c ˜ Ž ' V0WY  c i ®  µ .- :     = C:dc eV0WY .? i X#V0WY  :dc i ® # ?Qµ .-  - ›  = ® # 1  V0WY .? X#V0WY  i :  C:%c :dc i ë  ® .  ?  .  ?  = VXW/Y .? 0 V  W L Y ' X V / W Y i   X V / W Y #  # 1 # = = Ž i i i i óó ó Ö óó ó Ö Üƒ ×7 Ü 7 ð Î Ï Ø ; Î Ð terms ó7 ó contain ó Ö7 integrals of the type, ˜ Ü ×7 ƒ Theóvarious  2 ð ° Î Ï ØC; ;b* s Î Ð Ø Ï  Î Ð Ø ÑXÓ’Ø Ï ØC; s Î Ð Ø Ï  Î ˜ƒ 7 7 7 Î Ï ØC; ÎÒÑXÓ Ï s Î Ð Ñ Ï  Î ÑXÓ’Ø Ï Ø%*X; ÎÒÑXÓ Ï s Î ð ° Î Ï ˜ ˜

3

§

i

‘’

­ 7 8 ­ ®~¯ ®~¯ ¯ 7 ´µ¶ 8

ª

3

  l 

ª

„} } ;« ª

ª

ª

° ²± ° 7 8 · ° ©± °

 

%­ ®~¯

.?

i

ª 

ª

³°

°

³° ²±· °

°

©± °

Ü

Ñ Ï ; 2 s Î Ð Ø Ï  Î Ð Ñ ÑXÓ’Ø Ï , × Ü ÑXÓ’Ø Ï , and  Ø XÑ Ó’Ø Ï . Ð ÑÏ Î

©± °

ª

174

L. Fishman, M.V. de Hoop & M.J.N. van Stralen ® Ž

where

¬

   =

?

d

VXW/Y .?

i

ª

###*g

i

l   =

V0WY .?

i

ª #

G

®

 -   ™’™’™  :

(129)

µ .-   £ ® #  _Z¾ ¶ 42 3 * X#V0WY   :  :dc :dc i c i :,¿ c  £># 1 V  >£  ® ?  V  £>##* ?  £>  Ž V0WYL' i ë® £ ? i (130) O [ TS    º = º  c^C: . An asymptotic evaluation of 6 in the ¶ and limit can then, for the most part, be reduced to a stationary  £>

? V0WY \ ± 2

 Y



  



 

phase evaluation (38, 39) of the first three integrals in Eqs.(129)-(130) in conjunction with a Laplace method evaluation (38, 39) of the semicircular ( -)integralO contributions in Eq.(129). [ TS   6 is found to be The principal part of

' ë ®? ¶



 1 243 *

ª  €) 9

˜



* ?ঠ* * ?S * 243 * z 6

hence, the analysis naturally divides into two cases: (1) evanescent regime).

®

® ç

 ç



(locally-propagating regime) and (2)

 ç

® ç



(locally-





ç . Applying the stationary phase and the Laplace methods to the operator symbol integral representations given Case 1: ç in Eqs.(129) and (130), the dominant contributions are found to result from exterior and interior end points, interior critical points, and the singular points of the integrand. Since the Cauchy integral theorem (35) implies that the representation is independent of  º = º = = c^C: , this can be exploited in the detailed calculation. The principal idea is to choose so the particular choice of for that (i) the dominant contributions can be divided into three disjoint groups which can be calculated independently and added in the end to produce the final result, and (ii) the contribution from the singular points of the integrand can be evaluated in an expeditious manner. £¼  The first group comprises the contributions from the exterior end points at and :dc . The second group comprises the contributions from the interior critical points. The series and integral representations of the Lerch transcendental function, given, respectively, in Eq.(82) and Eq.(83), readily establish that, in the context of the high-frequency asymptotic analysis, in the function ,

¬ ¸

¸B¹µ .- C:   £ C:%c £ § ' 

µ .-

X#VXW/Y 

1 %: c : X#V0WY 

 £ : C:dc :dc µ .-   £ X#V0WY : :dc and

i



:dc i

® #

is solely an amplitude function, while

™G

® # G :%c i

® #

is finite except at

£gG

(131) .

¬ Thus, the oscillatory part of is the exponential, V0WYL' ë® £ ? i {   £># 1  i  i£ ˜ say, associated with its phase, follow ; 2 which implicitly contains the ‘large’ parameter ¶ . The (‘interior’) critical points, ˜ ® ?  V  £## *   ? £ ? £ £ j£ ˜ ,c ˜ , and :dc ˜ , where from the equality , and are given by ˜ , c  ë  ®  ? £˜  (132) »º    \ '   ë®  1"?¹243 *%- _ Á'  1 243 *   2 |l2¼ ^ i ? ' ë®    ?¹(133) 1 243 * * i i   ` ë® š  £ with the principal branches understood. (Observe, that as    , then ˜ c ch: and the critical points tend to the isolated ë® - £ „  E   singularities, while as    , ˜ and the critical points tend to the exterior end points, and :dc , and to c .) £g £g = = The third group comprises the contributions from the four interior end points at in conjunction c^C:x½ and c^C:x½ £g £g = with the two semicircular (ª -)integrals deriving from the -neighborhoods of the isolated singular points at c^C: and c^C: .

With the exception of the two previously mentioned limits of coalescing critical points, the contributions to the high-frequency asymptotic operator symbol evaluation from the three groups are disjoint if there are no critical points on the intervals on the ? =  ? =  ® ? = '  =d1 '  =d1 £ º ch: ch: c^C: imaginary -axis, c^C: and c^C: . Thus, for fixed ¶ and ¶ , to ensure that ˜ , requires the condition,

i

®





º





   = ##*›m

(134)

On the other hand, the radius £ singularities when ˜ Lc^C: .

c

=

of the semicircles must be sufficiently large to avoid a residual contribution from the isolated

Exact square-root operator symbols

‘’

175

=

The further refinement of the contour in relation to the calculation of the contributions from the -neighborhoods of the two isolated singularities requires some attention. The exponential occurring in the integrand of the semicircular contributions to Eq.(129) can be written in the form

V0WYL' û® = V0WY .? i

where

;¾  ª

and

¿ ª

ª

 ª

ì ® =

l % ª

  l   =

,

9 ® = }  ,

N N

i

I

V0WY .?

i

ª

2 * i

 '  l   =

V0WY .?

2

 '  l   =

V0WY .?

*

;¾  ª

## 1 Á  V0WYi' N

i

l   =

V0WY 

ª #I\l   =

V0WY 

# ?

ª

i

I

i

i

ª

i

¿  ª N

1 

# 1

ª

(135)

# 1

(136)

¾ ª

§ '  1 N   c . The estimation of the contributions from the two semicircular ( -)integrals in Eq.(129) is governed by with in N Eq.(135). The function (cf. Eq.(135)) can be written in the form }  = }  # N  ì ® = }  I m : % = % # ?  = }  # : : § '   1 N   N  ¥› N  Q  It follows that is a symmetric function of with respect to ch: for . For c . Also, note that c § '  N    = sufficiently small , it is straightforward to establish that is monotonically decreasing for c^C: and monotonically §(  1 = increasing for c^C: c with a global minimum at c^C: , transitioning for increasing to a ‘double-well-like’ structure with N B  ¥Þ a local or global maximum at c^C: and two, symmetrically located global minima. Condition (134), while ensuring that ? N B   º  §j   N   º  N   º  c c . Requiring that c^C: , is not sufficient to imply that using Eq.(135), resulting in the

ª

¾

l ª

 ª  l ;¾ ª ª

¾ ª

¾

 ª l

 ª

ª

¹

¾ ª

º = ; 2

 

l 

 = M N

¾ ª

 º

ª

;¾ ª

ª

¹ 

;¾

;¾

¾

¾

ª

additional condition,

®

ª

ª

(137)

§

    c implies that the exponential function in the semicircular

ensures the latter condition. Ensuring that contributions to Eq.(129) is dominated by the exponential decay in the high-frequency limit. Condition (137), however, is not N   N BB    sufficient to ensure that has a single, global minimum at c^C: . Requiring that c^C: ç ensures the latter condition, = resulting in the additional and final condition on ,

¾ ª

®



 

º





 = ##*  = : N  

;¾ ª Ensuring that

l 

¾

 =  ?¹- @m

(138)

has a single, global minimum at ch: not only ensures that the integrands corresponding to the contributions §j   from the two isolated singular points are exponentially small for c , but that, in addition, the dominant contributions in the G Laplace method (38, 39) calculation come entirely from the neighborhoods of the maxima at the end points and c . ® = In summary, the contour is specifically constructed such that for fixed ¶ and ¶ , a sufficiently small is chosen so as to satisfy the infimum of inequalities (134), (137) and (138), which simply yields (the subsequent inequalities hold for sufficiently = small )

ª

‘’

®





º



 

 = ##*  = : =

l 



ª

 =  ?¹-  º = ; 2

l 

 =  ºG

   = ##*›m

(139)

The choice of in Eq.(139) ensures that (i) the dominant contributions to the operator symbol integral representation given in Eqs.(129) and (130), with the general method of stationary phase (38, 39), can be divided into the following three disjoint groups: (a) (b) (c) = the

£g

and :%c , the exterior end points at ? £ ? £ £ ˜ , c j£ ˜ , and :dc ˜ , and the four interior critical points at ˜ , c £& = = the four interior end points at and ch: in conjunction with the two semicircular ( -)integrals deriving from c^C: £g -neighborhoods of the singular points at c C: and ch: , ^

›½

S½

ª

and (ii) the contribution from the two isolated singular points can be reduced to a Laplace method end point calculation. (a). Application of Eq.(84) and exploitation of the periodicity of the trigonometric functions,

 

 £,

:%c

9

 

 £>

and

V  £i

:%c

9

V  £#@

(140)

£g

reduce the exterior end point contributions from and :dc in Eqs.(129) and (130) to a single end point contribution from (of the semi-infinite integral representation, employing Eq.(81)),

YÀ„O Á  TS  6 xw [

? V0WY \ °±

c i

_Z¾ ¶ 243 * F K£e£0; 243 *IV0WYL' ë® £ ? i c›¿ ˜

{ 



 £># 1 V  £>

£g

176

L. Fishman, M.V. de Hoop & M.J.N. van Stralen ® Ž

  V  £>##*

?

?

i

  £>

(141)

Ã

(compare Eq.(79)). The standard evaluation of the end point contribution in Erd´elyi (38, pp.52-56) applied to Eq.(141) then yields the algebraic branch of the asymptotic operator symbol expansion,

YÀ„O Á  TS

¨w

  ‚ ˜ ' ë® 6 ¶

[

?



Ä2 ¶ *

 1"243 *g

 ® b  ' ë® ¶ ¶

?

l 

œ

 1 ; 3 *g|ëm /m mßm



Å

.- 

Å

(142)

 * 

The first two (non-vanishing) terms in the algebraic branch of the asymptotic operator symbol expansion are and ¶ ,   respectively, in contrast to the standard operator symbol expansion which contains an ¶ term (20). This difference is a reflection of the symmetry inherent in the Weyl construction (16, 17). Equation (142) can also directly be derived from the analytic continuation of the corresponding asymptotic result for the defocusing quadratic profile presented in Fishman (23, (27)) through the relationship given in Eq.(113). The algebraic branch coincides with the outcome of the polyhomogeneous calculus of operator symbols (1, 15, 16).

Å

(b). The stationary phase evaluation of the contributions from the four interior critical points, accounting for Eq.(131), is accom£ ¹£ ˜ plished in the standard manner as in Erd´elyi (38, pp.52-56) . The contributions from ˜ and c are combined as well as the ? £ ? £ ˜ and :%c ˜ . While carrying out these combinations, the following identity, ones from c

Ã

?

:

© :  : •

° 3 * µ 

 ?

œ

g

œ

© C:  •

µ 

 -

©

 *  ?¥  “ the residues in Eq.(149), are exact. This follows on starting with to show – in Eq.(152) and using Eq.(84) with “ that ? ' µ  .?L-

   ?¹-  ? µ  . ?L-  ?é # 1 : /C: C: : /: i - ?é  µ  .?L-   ? µ  . ?L-  ] #] C: /: C: /C: i

é

é

é

establishing that Eq.(152) is valid for



|ì



é

ë

§ '  1 ¼ : with

é

é

for and

ê

ƒ

§j.-  1  :

é 

-

(155)

. Then, use of the identity

 & •  µ   •  9 µ/ ?-  •  M œ œ œ

(156)

TVUW

which O follows directly from Eq.(82), application of Eq.(83), and comparison with © [   >¥  – in Eq.(61) for æ ¸ complete the proof. The Lerch functional equation (151) makes explicit the number of “ propagating modes ¸ in the operator symbol, contour-integral representation in Eq.(102).



®



ç . For the high-frequency asymptotic evaluation of the operator symbol integral representation in Eqs.(129) and Case 2: ç £@( (130) in this case, there are (i) exterior end point contributions from and :%c , (ii) no interior critical points on the imaginary -axis, following from Eq.(132), and (iii) contributions of exponentially small order from group (c), following from Eq.(139) and the discussion preceding Eq.(145). Returning to Eq.(102) and the original contour , a critical point is encountered on the real ® ?  V  l# *  -axis: it follows from the equality , and is given by

i

i˜

º   

2 |l2¼í^ *  



   i

i

\ ' -Þ?aë®

 1"243 *’_



 -  ' -Þ?éë®  - ? ' -Þ?éë® ] 





 1 42 3 *  1 243 *

(157)



`

(158)

with the principal branches taken. The contribution from this critical point is a term of exponentially small order, supplementing the algebraic branch deriving from group (a). The most direct way to derive the resulting expansion is to exploit the analytic continuation result in Eq.(113) in conjunction with the corresponding asymptotic result for the defocusing quadratic profile, derived by a stationary phase evaluation, in Fishman (23, (27),(28)). The final expression is

[

O

¨w

TS   ‚ ˜ ' ë®G?  1 243 *  2 *  ® I  ' ë®? 1; 3* 6 ¶ ¶ ¶ ± ¶ ;b° 3 *  X; 243 *' -]?éë®  2  1 ; 42 3 V0 W Y^' ® ˜ ? ' -]?éë® ¶   * ¶ ˜

i





Ä

i







œ



l 

 1"243 * 1 Z|ëm m mDm

(159)

The principal results of Section 5 are the low-frequency asymptotic expansion in Eq.(128) and the high-frequency asymptotic expansions in Eqs.(146), (150), (152), and (159). Taken together, Eqs.(146) and (159) provide the nonuniform, high-frequency S asymptotic, operator symbol expansion for the full range of the phase space variables and 6 . As previously discussed, the high® š frequency results are not valid in the  limit due to the coalescing pairs of interior critical points. The high-frequency ® £ results will also fail in the limit of  due to the coalescing of (i) the interior critical point at ˜ and the exterior end point at ? £ ? £ £g g £  ˜ and the exterior end point at ˜ :%c , and (iii) the two interior critical points at c , (ii) the interior critical point at :%c é£ ˜ £ c . To address these two limiting cases, uniform methods must be applied (39). The nonuniform expansions, and c at however, do establish and illustrate the fundamental oscillatory character of the square-root Helmholtz operator symbol in the highfrequency limit (1, 23). In the elliptic pseudodifferential operator calculus, only the nonuniform algebraic branch given in Eq.(142) is obtained in the asymptotic analysis (1, 14, 15, 16, 17).

 

 

Exact square-root operator symbols

179

6. One-way propagation A. Propagating and nonpropagating operator symbol constituents It is clear from the derivation of the spectral (modal) summation representations presented in Section 2 that the Helmholtz operator symbols naturally divide into their propagating and nonpropagating modal contributions. In the Weyl calculus, in both Eq.(61) with  æ ¸ and Eq.(64), the first, finite sum of ¸ terms represents the propagating modal contribution to the square-root Helmholtz operator symbol, while the remaining terms comprise the nonpropagating modal contribution, with a corresponding decomposition  O FromO the propagating O in both Eq.(33) with æ and nonpropagating modal decomposition, ¸ and Eq.(48) in the standard calculus. [  [  [ adopting an obvious superscript notation, it then follows that with

î

O

[

TS  ÿ   6 TS  ÿ 6

ð;î O [

ïî rð;î

O

ª V [

TS    6 – TS   m 6 – O

© “ [ i “

(160) (161)

%Ÿ î

Ÿ ð;î

7O into their 7O real and Correspondingly, decomposing the previously mentioned propagating and nonpropagating modal contributions 7O S N  N N for the imaginary parts, and further noting the even/odd symmetry with respect to (or 6 ) establish that standard symbol, with

%Ÿ î 7O

N

 V0WY .? S  i 6

%Ÿ ð;î

and

 ñRò



7O

ª V V0WY  S N i 6 “

TS   @ 6 – i

ó

7O ª V V0WY  S  N TS   ] i 6 i 6 – “   where the even and odd parts of a function 5 are defined by .?  1  '   1  2 '  , 5 5 5 * 7O

N

ÁV0WY .? S  i 6

z

ñRò

z

and

ó 'z ò

 1  5 *

†z

z

z

2 '   ? 5

.?

ó

  ò

5



z

©

ò

“ ©

ñ(ò

“

7O

V0WY  S N i 6 V0WY  S  N i 6

7O

TS   6 –

£

TS   6 –

£ 



(162)

(163)

(164)

1 

(165)

S

respectively, and are taken with respect to either or 6 in Eqs.(162) and (163). Equations (160)-(163) can be applied to any exact representation of the square-root Helmholtz operator symbol. In particular, equating the Weyl operator symbol constructions (61) and (102) in Sections 2 and 3, respectively, yields

î O

[

; 2 TS  9 - ?®  1"243 * 42 3 * «¥‰ ' ?„ ¬  ] 6 d: ¶ : ˜

!

or

î O

[

TS  9   ª V 6 Ž V0WY¼' ®

and

[

ð;î O

TS  g   6

or

[

ð;î O

?

i

? VXW/Y \ ± 2

c i

 

O 

F 8K

J I 

? V0WY \ ± 2

c i

_j¾ ¶

243 * :,¿

i   i   bi S ® ?

 b 1 V

c ?

µ .-

:

 .?

i C:%c

i

 #V0WY 

  V  bi ##* ?     bi C›

?

ƒ ¾ ¶ 243 * F K£e£X; 243 *IV0WY ë® £# c ¿ ˜

A

(166)

 ui

_Z¾ ¶ 243 * :^¿ c

i ?     bi  1 V  bi S ®

TS  &   © 6 i Ž V0WY¼' ®

A

«

F

I

 £> ? @ £’ï  ¸

 uK i

µ .-

:

:%c i

® #

G

®

 -   ™ž™0™  :

L

 .?

(167)

(168)

i C:%c

i

 #V0WY 

  V  bi ##* ?     bi C› ®

:%c i

G

® #

 -   ™ž™0™  :

(169)

with analogous results following from the application of Eq.(64). Following from Eqs.(166)-(169) and the constructions presented in Section 5, the high- and low-frequency, asymptotic expansions for the propagating and nonpropagating constituents of the square-root Helmholtz operator symbol can be obtained. In the

180

L. Fishman, M.V. de Hoop & M.J.N. van Stralen

™ î TS  6 ¡¨ ‚w ˜ õ öô

high-frequency, ¶

O [

 ª V

, limit, for example, utilizing Eqs.(146) and (159) result in



®





rhs of Eq.(146) –

“

ª V “

rhs of Eq.(159) –

k

 ç ®

ç

ç

(170)

k ç

with a corresponding set of expressions for the nonpropagating operator symbol constituent. The decomposition into the propagating and nonpropagating constituents in conjunction with the high-frequency asymptotic expansions relate the number of propagating modes ¸ to specific structural features of the square-root Helmholtz operator symbol. The oscillatory asymptotic branch in Eqs.(144) and (146) is governed by the exponential phase functions, ® S * ¦ * * ' ë®  ?a- 1 243 * _ \® £˜ ? . Taking  and  6 in Eqs.(132) and (144) then yield the dominant exponential phase … … … function (to within a sign),

½

\® £˜ ?



p÷





' ë®

 ?¹- 1 243 * _

 

ø 3 wùúk2

\ ®Á

÷

c^C:

 ? :

® 243 *



243 * _

á

ú w] ž

\ ®Á

ch:

 ?

® 243 * ; 243 * S _@m ¶ :

Thus, in the lowest order,

/

®

 á #* ¶ ^ c  á

û

»û

O

[

(171)

î

S

is the period of oscillation of as a function of . An estimate of the number of propagating modes ¸ follows where immediately from the estimate following Eq.(19), with more sophisticated expressions following in a natural manner. ) * Moreover, the addition of one propagating mode through the increase in ˜ and/or decrease in ¶ , mathematically represented ®   - ® ¸ : O ), corresponds to a change in the sign of the scaled, nonpropagating, square-root Helmholtz by taking ¸ ( ©  [  >C ; 243 * –Þ¶ operator symbol constituent (i ) in Eq.(152) in the high-frequency limit.

‹



“

B. The one-way propagator The infinitesimal, one-way propagator, in conjunction with the special structure of the Weyl and standard symbols for the squareroot Helmholtz operator, reveal both the multiresolution and generalized screen nature of the propagation process. The fundamental    !/z B  ! B 9 {  ? B     !/z B  ! B     !/z B  ! B  solution of the one-way wave equation can be written in the form y with 5 5 5 5 5 5 5 5 ”  denoting the one-way propagator. Following from Eq.(10) with , the infinitesimal, one-way propagator follows as

s

! z B  ! B 9    5 5

s

! z   / 5 5 

i

 :dc

.K S VXW/Y



i

 

O

Sh"!?
\ S  2 "!  ! B  _  ? B  5 5 *

£

m

(172)

B 5 , the propagator can be asymptotically expanded to the lowest order in the form B! B = "!?
.K S V0WY Sh"!8?! B  [ \ S  2 "!  ! B #_ 5 5  :dc i *

In the limit 5

s

E

F HJ 

s

(w






(173)

A "!  ! B 

where the integral term is recognized as the Schwartz kernel (cf. Eqs.(6) and (8)) and is understood in the distributional O sense (4, 30). [ TS   Recognizing from Eq.(63) that the Weyl operator symbol 6 can be written as the sum of an absolutely and uniformly š ïS ï  convergent infinite series and a well-defined integral, it follows that in the limitO , the series contribution tends to [ TS   ïS ï D   O while the integral contribution reproduces the pseudodifferential operator limit i . In the limit , but , 6 [ TS   discarding the integral contribution and approximating 6 by the infinite series alone result in a convergent integral in Eq.(173), corresponding to an essentially correct treatment of all of the modal contributions except the ‘large ¬ ’ values in the    !z B  !CB     !z B  !CB  (generally) deep evanescent regime. Denoting this approximation by , the expansion in Eq.(173) 5 5 5 5 takes the form

 {w

s A

(w

'?

ƒ

 ? B  «¥‰ 5 : i 5

= "!?
  -]? 



ýG

xw s 

s

! z B! B   / 5 5

üE

(174)

Exact square-root operator symbols

181

; 2 \ ± 2 ¦ * "! «  ! Œ B  *  S * _ . The last line of this equation can be written in the form « ‹~˜  M FIH J 

.K S„.?   V0WY .?  VXW/Y,' Sh"!8?
where

èþ 

¶





S !  .K S V0WY . ? ; 2 S *  V0WY  b  C:%c « « Œ ¶ : i  V0WY ? ± 2 ; 2 ¦ * " !  ! B #* ^V0WYi' Sh"!  ! B  1 m Ž .?  ¸ ‹~˜  : M ¶ i

 F HJ 

p





(175)

In Eqs.(173)-(175), a forward constituent transform is identified as the windowed Fourier transform

-

ÿ

: [



:

¦

&QV0WY # ?

"!  ! B  *

[

#?

FbH JLK ! B VXW/Y

:

ÿ

S

i

:

"!  ! B 

ÿ

:

&



ÿ : 7 HJ

where  is a dilation parameter, with a corresponding constituent ‘inverse’ transform, multiresolution (40) nature of the propagation process. « The standard calculus analogue of the (modal) expansion in Eq.(174) is given by

(w

s

! z B! B   / 5 5

{A Ž

'?

ƒ

 ? B  243 * «¥‰ i 5 5 :

= "!?


 ¾  -Þ? Thus, the , subjected to the appropriate normalization, form an orthonormal basis. With Eq.(177), in the action of the propa! B gator, the integration over constitutes a ‘forward’ transform, whereas the summation over ¬ constitutes the associated ‘inverse’ transform; the propagator is ‘diagonal’ in the transform domain. Unlike the global operator diagonalization in the spectral (modal) propagator representation, the infinitesimal propagator representations can be viewed as a ‘diagonalization’ in the phase space strip about a localized coordinate point (24).

7. Exact symbols for operator rational approximations to The square-root Helmholtz operator can be approximated by operator rational approximations in general, and additive, operator rational approximations in particular (1, 3, 4, 5, 6, 10, 23). These operator approximations implicitly correspond to uniform operator

182

L. Fishman, M.V. de Hoop & M.J.N. van Stralen

symbol constructions over appropriate regions of phase space, and immediately result in approximate, partial differential, one-way  wave equations (1, 3, 4, 5, 6, 10, 23). With a continued-fraction approach, the square-root Helmholtz operator is approximated to æ th-order by




¨'  where

1"243 *



/



  ¤ ã 

 ã‰

' 

 2




 ¤ã 

1; 2 



(178)




* ?  ('  ) * "!  ?-  k .- l

 * $* 1   

(179)

 ¤  ¤

the operator sum in Eq.(178) is supplemented with a right-traveling- (outgoing-)wave radiation condition, and the approximation   coefficients  ã 㠖 are determined in the homogeneous medium limit by a variety of approximation-theoretic criteria (1, 3, “ 42). For every case where an exact square-root Helmholtz operator symbol can be constructed, the operator symbols P R 4, 5, 6, 10, 23, TS   [  corresponding exactly to the operator sum in Eq.(178) can be written in closed form (1, 23). Let the Weyl symbol 6 '  1"; 2 correspond with the operator , then Eq.(178) can be simplified as

J

/  +
 


    ¤ ã '
? '
  ¤ ã 

 ã‰



 2

   É

1 ; 2#1 

so that the exact, additive, rational approximation operator symbol in the Weyl calculus is given by (1, 23)

J

  O

[

where

  ¤ã

  ¤¤ ãã



  n    P R TS

 ¤

-   ㉠ ' -Þ? [ 㠍 2

TS  9   6

 1  6

(180)

m

    P R TS É

[

(181)

  6 is the fundamental function in the construction. It follows from (i) the relevant results

The operator symbol OQP R in Fishman (23) for the defocusing quadratic profile in conjunction with the analytic continuation result in Eq.(112) and (ii) the TS   [ previous construction of 6 for the focusing quadratic profile in Eq.(56) that

    É [



P R

TS  &   6 ¶

:



ƒ «¥‰



«

˜

' ?„ ¬  -]? ®  1 ; 2 :

 

O 

(182)

   - É ®  V0WY  :dc i



P R from (iii) the observation that the appropriate Lerch transcendental function for the spectral (modal) summation representation, and TS   µ   •   [ G  6 appears in the analogue of Eq.(92) with œ involved in the construction of , that [

 !  P R TS É

 &   6

-

? ¶



-Q?

(183)

G   -  :  ™ž™’™ Ž uK iV0WYKJ ®  i ? {    bi  L V  bi M ®  

Ž F



for the contour-integral representation. In Eqs.(182) and (183),

®



 ® 

¶

; 2 '

  ,

?a- 1 m

 !  P R TS Appendix B contains an additional representation for É [

(184)

  6 . Combining Eqs.(182) and (183) with Eq.(180) results in

the exact, closed-form representations of the Weyl symbol for the additive, rational operator approximations of the square-root Helmholtz operator. Finally, expressions analogous to Eqs.(182), (183), and (B3), in conjunction with Eqs.(180)-(181), can be written for the operator symbols in the standard calculus in a straightforward manner.

8. Numerical results The square-root Helmholtz operator symbol for the focusing quadratic profile can be numerically computed, in both the Weyl and standard calculi, from the spectral (modal) summation and contour-integral formulas. For the spectral (modal) summation

Exact square-root operator symbols

Figure 7. v Eq.(104).

Figure 8. v Eq.(104).

7O

7O

Ü ÛCÝ Î÷

Ü ÛCÝ Î÷

" Ï vs. ÷

for the focusing quadratic profile. The exact standard operator symbol is computed from

" Ï vs. ÷

for the focusing quadratic profile. The exact standard operator symbol is computed from

183

representations, for the standard operator symbol, for example, Eq.(48) is computed. The finite and infinite sums are rewritten and evaluated in the manner outlined in detail by Van Stralen (4), while the remaining finite integral is computed by an adaptive recursive Newton-Cotes 8 panel rule (4, 43), with chosen relative to the magnitude of the integrand in a manner which balances the location of the second, infinite series within its circle of convergence against the magnitude of the integrand and the range of integration in  that final term. In this scheme, it immediately follows from the estimates in Eq.(42) and Eq.(44) that the limit corresponds to the infinite sum approaching its radius of convergence, requiring an ever increasing number of terms for an accurate numerical š computation, and a relatively simple numerical integration, while the limit corresponds to the sum approaching the center of its circle of convergence, requiring an ever decreasing number of terms for an accurate numerical computation, and a more involved numerical integration (44). The expression in Eq.(48) is independent of the particular choice of , which was verified in the numerical computations. The corresponding computation in the Weyl calculus of Eq.(64) is treated in a similar fashion. For the



 



 

184

L. Fishman, M.V. de Hoop & M.J.N. van Stralen

Ü ÛCÝ

Figure 9. v Eq.(104).

7O

Î÷

" Ï vs. ÷

for the focusing quadratic profile. The exact standard operator symbol is computed from

i

contour-integral representations, the Weyl and standard operator symbols are computed from Eq.(102) and Eq.(104), respectively. ª BB in Fig. 6 with taken to be finite. Applying Eq.(84) and In both cases, the contour in the complex -plane is chosen as exploiting the periodicity of the hyperbolic functions in Eq.(107) in the same manner used in proceeding from Eq.(106) to Eq.(108) BB BB reduce the contour integral to a single integral along 2 and two integrals along * . The Lerch transcendental function is computed ª ªD® from Eq.(83) by the Romberg numerical integration method (45) applied to the three resulting integrals. is chosen to make ª .-    an quantity, thereby balancing the numerical effects of the singularities at i ch: and i ch: for the limit with the ª š .-  attempt to construct an quantity from the integration of extremely large magnitude integrands in the limit . In both the ª Weyl and standard cases, the computations are, in principle, independent of the choice of , which was verified numerically for a reasonable range of the parameter. In the subsequent numerical computations, the spectral (modal) summation and contour-integral ® º¨methods resulted in identical curves. Further, for the case , these two computational methods were found to be in complete   agreement with the results obtained from numerically integrating Eq.(125) and its standard calculus analog, Eq.(33), with æ . The exact operator symbol curves presented in this section were computed from the contour-integral representations. 7O TS N   6 is (i) invariant under It follows from Eqs.(48) and (104) that the standard, square-root Helmholtz operator symbol S ¦ S S è S S è 6 , (ii) a symmetric function of (6 ) for 6 ( ) the interchange , (iii) an asymmetric function of (6 ) for 6 ( ) , S ® º and (iv) a (an) symmetric (antisymmetric) function of and 6 for the imaginary (real) part of the symbol for in the 7O TS N > m  absence of propagating modes. The third and fourth points are illustrated in Figs. 7-10 by plotting for the specific ) *    -¼? *

 m - m m  m

 m quadratic case 6 6 and  , , , and , respectively. Figure 10 for  illustrates the fourth point. The sequence of figures also illustrates the transition from the high- to the low-frequency regime for a choice of q within the well, with the square-root function plus superimposed oscillatory behavior, characteristic of the locally-homogeneous, highfrequency limit (1, 4, 23), gradually transforming to the absorption-dominated curves, corresponding to the absence of propagating 7O N TS  - m 

a -  m modes, in the low-frequency limit. Figure 11 displays for the same , illustrating the dominant 7O  profile and  N   

ì -  m absorptive behavior for a choice of q outside the well. Figure 12 illustrates , and will be 6 for the same profile and  applied in demonstrating the waveguiding properties of the focusing quadratic profile in the final Sec. ??. All of the Figs. 7-12 are consistent with the appropriate analytic continuation of the corresponding results for the defocusing quadratic profile presented by O Fishman (23). [ TS   6 is (i) solely a function It follows from Eqs.(64) and (102) that the Weyl, square-root Helmholtz operator symbol of the variable , following from the symplectic structure in the Weyl composition equation (11) and the quadratic dependence of S ¦ S ) *   O 6 (16, 23), (ii) invariant under the interchange 6 , (iii) a symmetric function of and 6 , and (iv) purely imaginary for ®èº [ TS >C ) *   -8? * 6 6 in the absence of propagating modes. Figures 13-17 illustrate for the specific quadratic case

Á m -  m m m m and  , , , , and , respectively. As in the case for the standard operator symbol, the sequence of figures again illustrates the characteristic behavior and transition from the high- to the low-frequency regime for a choice of q within the



Å

j

k

k

i

Å





#

G

4 4

4

4

4

4



4 4

4

$#

4

%4

4

4

4

4

Exact square-root operator symbols

7O

Figure 10. v from Eq.(104).

7O

Figure 11. v from Eq.(104).

Ü4ÛCÝ Î÷

Ü Ý Î÷ Ñ

"Ï

vs.

÷ for the focusing quadratic profile. The exact standard operator symbol is computed

"Ï

vs.

÷ for the focusing quadratic profile. The exact standard operator symbol is computed

185

well. Analogous behavior to that displayed for the standard operator symbol in Fig. 11, for a choice of 6 outside the well, follows immediately from Fig. 14 and the variable dependence on . Once again, all of the Figs. 13-17 are consistent with the appropriate analytic continuation of the corresponding results for the defocusing quadratic profile presented by Fishman (23). The differences in the symmetry properties between the standard and the Weyl, Helmholtz operator symbols, exhibited in the formulas and illustrated in the preceding figures, are a reflection of the different operator-ordering schemes underlying the two pseudodifferential operator calculi (17). The effectiveness of both the low- and high-frequency asymptotic, Helmholtz operator symbol expansions derived in Sec. ?? is readily demonstrated. Figures 16 and 17 compare the exact result in Eq.(102) and the low-frequency asymptotic result in Eq.(128) -  š  - m : and , respectively, suggesting the increasing accuracy as ¶ for ¶ and the manner of breakdown of Eq.(128). The ®èº»same results are obtained using Eq.(125) for the exact operator symbol calculation for . In Figs. 13-15, the exact result in



4 1

™

186

L. Fishman, M.V. de Hoop & M.J.N. van Stralen

Figure 12. v Eq.(104).

7O

ÛCÜ.ø Î

ø Ï vs. for the focusing quadratic profile. The exact standard operator symbol is computed from

)K *(q+) = 1 − q ,k = -50.5 Exact Re Ω( Exact Im Ω( HF Re Ω HF Im Ω(

2.0

2

1.5

2

B B

B

Ω B (p , 0)

B

1.0

&0.5 &0.0 -0.5

&0.0

0.5

1.0

.0/21 3 46587 3

'p

1.5

2.0

Figure 13. vs. for the focusing quadratic profile. The exact Weyl operator symbol (Eq.(102)) is compared with the Weyl, high-frequency (HF) approximate, operator symbol (Eqs.(146) and (159)).

;9 := FBk

9

Exact square-root operator symbols

v2.0

y* +

y

K 2 (q ) = 1 − q 2 1.5

wExact Ω wExact Re Im Ω( HF x Re Ω(

187

k = 10.5 B B

Ω B (p , 0)

B

HF Im Ω B

1.0

u0.5 u0.0 -0.5

u0.0

u0.5

1.0

'p

. / 1 3 46587 3

1.5

2.0

Figure 14. vs. for the focusing quadratic profile. The exact Weyl operator symbol (Eq.(102)) is compared with the Weyl, high-frequency (HF) approximate, operator symbol (Eqs.(146) and (159)).

}~ 

2.0

1.5

Ω B (p , 0)

}

€

K 2 (q ) = 1 − q 2 k = 3.5 Exact Re Ω B Exact Im Ω B HF Re Ω B HF Im Ω B

|

|

1.0

0.5

0.0

-0.5

z0.0

. / 1 3 46587 3

z0.5

1.0

{p

1.5

2.0

Figure 15. vs. for the focusing quadratic profile. The exact Weyl operator symbol (Eq.(102)) is compared with the Weyl, high-frequency (HF) approximate, operator symbol (Eqs.(146) and (159)).

9 MV< , limit, the Weyl composition equation (11) can be approximated by W /‚ X RƒZ ` \„: ] ^ Xa` \c…R ^O† h W /rX RƒZ ` \6k ^ Z (185) ^ which can serve as the basis for an approximate, high-frequency reconstruction (23). This is illustrated in Fig. 19 where W / X F (9Š:g GCHFDI ). The (computed from Eq.(102)) is compared with accuracy of the reconstruction increases as 9‹Mb< , with the deviation from zero imaginary part serving, in some sense, as a measure In the high-frequency,

of the accuracy of the profile reconstruction for real profiles (23).

188

L. Fishman, M.V. de Hoop & M.J.N. van Stralen 2.0

}~ 

z

k = 0.95

|

|

Exact Re Ω B Exact Im Ω B LF Re Ω B LF Im Ω B

|

1.0 Ω B (p , 0)

Œ}

K 2 (q ) = 1 − q 2

1.5

0.5 0.0 -0.5 -1.0

z0.0

z0.5

1.0

1.5

2.0

p

. / 1 3 46587 3

Figure 16. vs. for the focusing quadratic profile. The exact Weyl operator symbol (Eq.(102)) is compared with the Weyl, low-frequency (LF) approximate, operator symbol (Eq.(128)).

~Π

3.0

| |

2.5 2.0 Ω B (p , 0)

z

Œ

K 2 (q ) = 1 − q 2 k = 0.1 Exact Re Ω B Exact Im Ω B LF Re Ω B LF Im Ω B

1.5 1.0 0.5 0.0 -0.5

z0.0

. / 1 3 46587 3

0.5

1.0

1.5

2.0

p

Figure 17. vs. for the focusing quadratic profile. The exact Weyl operator symbol (Eq.(102)) is compared with the Weyl, low-frequency (LF) approximate, operator symbol (Eq.(128)).

9. Discussion The Helmholtz operator symbols for the focusing quadratic profile have been exactly constructed by two complementary methods: (i) a spectral (modal) summation approach in Section 2 deriving from standard representations and constructions in spectral analysis and (ii) a contour-integral approach in Section 3 based on extracting the operator symbols from the appropriate Green’s function, or corresponding parabolic (Schr¨odinger) propagator, data. As the name implies, the former method is natural for partitioning the operator symbol into its propagating and nonpropagating components, as well as for examining the individual modal contributions, while the latter method is natural for examining the total operator symbol, in particular, for deriving both the high- and

Exact square-root operator symbols

}~ 

1.4

189

}

K 2 (q ) = 1 − q 2

1.0

Ω B (0 , 0)

0.6

Exact Re Ω

Im Ω| ŽŽExact HF Re Ω HF Im Ω|

0.2

B B

B

-0.2

B

-0.6 -1.0

z0.25

-1.4 0.00

0.50

0.75

1.00

1.25

1.50

1.75

2.00

k ‘ Figure 18. . / 15E4587 vs.  for the focusing quadratic profile. The exact Weyl operator symbol (Eq.(102)) is compared

with the Weyl, high-frequency (HF) approximate, operator symbol (Eq.(150)).

“

1.8

2

“ ” ”

1.2

“ ” ”

k = 50.5

|

2

Re Ω B2

|

“ •”

k = 50.5

0.6 Ω B2 (0 , q)

’K ~ ( Œ q  ) = 1 − Œ q

”

k = 10.5

Im Ω B2

k = 3.5

•”

k = 3.5

1-q

“

2

”

k = 10.5

0.0

-0.6

-1.2

z0.0

^ Figure 19. . / 1 5E4–ˆ7 vs. –

0.5

1.0

1.5

q

for the focusing quadratic profile. The exact focusing quadratic profile, is compared with the square of the exact Weyl operator symbol computed from Eq.(102).

— ^ 1˜–B7š™›œŠ– ^ ,

low-frequency asymptotic expansions. The two approaches can be combined, as in Section 6, to derive both integral representations and asymptotic expansions for the individual propagating and nonpropagating modal sums, as indicated in Eqs.(166)-(170). In particular, for the propagating contribution to the square-root Helmholtz operator symbol, in the high-frequency limit, a sum of a finite, but ever-increasing, number of terms (cf. Eq.(166)) is asymptotically analyzed in a straightforward manner through the contourintegral equivalence as indicated in Eq.(170). Furthermore, even though the periodicity of the associated parabolic (Schr¨odinger) propagator is explicitly exploited in the contour-integral construction and the discrete nature of the spectrum is, likewise, exploited in the spectral (modal) summation construction for the focusing quadratic case, the two complementary operator symbol construction procedures and their combined usage are quite general, and applicable to other profiles. Indeed, for the Helmholtz operator

190

L. Fishman, M.V. de Hoop & M.J.N. van Stralen

symbols in the defocusing quadratic case, which were constructed by the same procedures (4, 23), the associated spectrum has no discrete contributions and the associated parabolic (Schr¨odinger) propagator is not periodic. The fractional, Helmholtz operator symbols constructed in Sections 2 and 3 represent the appropriate Helmholtz operator roots associated with the physical, right-traveling wave field. In the spectral (modal) summation construction, this condition is enforced through the appropriate infinitesimal shifting of the resolvent singularities in the integral representation (1, 4, 23), while, in the contour-integral construction, the correspondence to the physical roots follows immediately from the extraction of the operator symbols from the physical, outgoing wave Green’s function or the corresponding parabolic (Schr¨odinger) propagator (23). As such, the Helmholtz operator symbols must satisfy the appropriate composition equations and be consistent with the physical, right-traveling- (outgoing-)wave radiation condition (1, 4, 23), as was demonstrated for the defocusing quadratic profile case by Fishman (23). This is briefly outlined for the focusing quadratic profile case in Appendix C. Satisfying the appropraite composition equations alone is not sufficient to ensure the construction of the physical, Helmholtz operator symbols; the correspondence with the radiation condition is essential. For example, if the i limit is taken in the analytic continuation formulas (112)-(115), the resulting symbols will still satisfy the appropriate composition equations (at least in a formal, asymptotic sense (15, 17)), however, they will no longer be consistent with the physical radiation condition, and, thus, will not correspond to the physical operator roots. The construction of the physical, Helmholtz operator symbols has been briefly discussed by Fishman (23). The exact, square-root Helmholtz operator symbol constructions given in Eqs.(102) and (104) in conjunction with the numerical results presented in Section 8 illustrate the waveguiding properties of the focusing quadratic profile. At the level of the marching range step, which follows immediately from Eq.(10), the infinitesimal, down-range wave field is given by (19, 20, 21, 23)

ž MQc ž

Ÿ‹  Xa¡r¢£Y¡ Z¤¥\ †§¦ ¨ © X ed KEG8ª\«R­¬¯®° X ied R ¤H\‚¬¯®° h ied ±²/ X RƒZ³¤¥\ £Y¡ k Ÿ ´   Xa¡ ZaR \µ> (186) Taking the initial wave field at ¡ to be represented by a very broad Gaussian function, which is essentially constant over the non] ^ absorptive range of the profile ( X ¤H\·¶¸< ), results in a very narrow Gaussian function, sharply peaked about R:¹< , for the Ÿ corresponding ´   Xa¡ ZR!\ , leading to the approximation Ÿ‹  Xa¡r¢£Y¡ Z¤¥\ † ¬¯®° h ied ±º²/ X (187) ± ² For the cases corresponding to propagating modes, it follows from Fig. 12 (and the R»¼ž ` invariance of / X RƒZ ` \ in the qua-

dratic case) that the energy within the effective waveguide will be redistributed, with the wave field in the absorptive regions being suppressed. In particular, the oscillatory character of the operator symbol ensures the strict conservation of the integrated energy flux (1, 19), while the phase space regions with ultimately allow for the down-range focusing correctly corresponding to the modal energy distribution. While the focusing quadratic profile is, in some respects, nonphysical, the corresponding Helmholtz operator symbols, nevertheless, establish canonical symbol features for more general profiles containing locally-quadratic wells. This should not be surprising. In the context of a modal analysis, the low-lying eigenfunctions and corresponding eigenvalues in such wells will exhibit a quadratic character. This is the same phenomenon expressed at the level of the operator symbol. This will be illustrated in detail elsewhere. The Helmholtz operator symbol results presented in Sections 2-7 can be immediately extended to the generalized, focusing quadratic profile defined by

no½p ± ²/ X RƒZ ` \[tY¾g


^ Xa` cÀ Kˆž ^ \ ^ ¢ R s^ Ç Z ¿ ^ ¢ À ^ KBž ^ Ç Z Xa` cÉÀ KBž ^ \R Ç Z

À­:§