On Trace Formulas for Schr odinger{Type Operators - Semantic Scholar

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On Trace Formulas for Schrodinger{Type Operators

F. Gesztesy H. Holden

Vienna, Preprint ESI 250 (1995)

Supported by Federal Ministry of Science and Research, Austria Available via anonymous ftp or gopher from FTP.ESI.AC.AT or via WWW, URL: http://www.esi.ac.at

August 23, 1995

ON TRACE FORMULAS FOR SCHRO DINGER-TYPE OPERATORS F. GESZTESY1 AND H. HOLDEN2

Abstract. We review a variety of recently obtained trace formulas for one- and multidimensional Schrodinger operators. Some of the results are extended to Sturm-Liouville and matrix-valued Schrodinger operators. Furthermore, we recall a set of trace formulas in one, two, and three dimensions related to point interactions as well as a new uniqueness result for three-dimensional Schrodinger operators with spherically symmetric potentials. 1. Introduction. It is a well-established fact by now that trace formulas are of great

importance in solving inverse spectral problems for Schrodinger operators. This is demonstrated in great detail in [7] in the context of short-range inverse scattering theory and in [9], [11], [26], [33], [38] in connections with the inverse periodic spectral problem. Historically these trace formulas originated in the works of Gelfand and Levitan [14] (see also [8], [12], [13]) for Schrodinger operators on a nite interval. Subsequent developments extended the range of validity of trace formulas in a variety of directions including algebro-geometric quasi-periodic nite gap potentials and certain classes of almost periodic potentials [6], [27], [30]{[32], [39]. Moreover, trace formulas proved to be a vital ingredient in descriptions of the isospectral manifold of quasi-periodic nite-gap potentials and some of their limiting cases as well as in the corresponding Cauchy problem for the Korteweg-de Vries equation. Due to the somewhat special nature of the potentials covered in the references cited thus far, it seemed natural to search for extensions of these trace formulas to a large class of potentials. This was the point of departure of our recent program which led to a trace formula for any continuous potential bound from below and subsequent generalizations to higher-order trace formulas in one dimension and certain multi-dimensional generalizations [15]{[19], [21]{[25],[37]. In the simplest case, the main new strategy is to compare the L (R) Schrodinger operators H = ? dxd + V and HyD = ? dxd + V , the corresponding operator with an additional Dirichlet boundary condition at the point y 2 R. The spectral characteristics of H and HyD , especially the Krein spectral shift function (; y) associated with the pair (HyD ; H ), then allows one to recover the potential V (y). In Section 2 we extend the results of [22] and [24] to Sturm-Liouville operators of the type r? [?(p f 0)0 + qf ] in L (R; r dx) and consider general self-adjoint boundary conditions 0(y)+ (y) = 0, 2 R in addition to the Dirichlet case = 1. Section 3 sketches an extension of the trace formula to matrix-valued Schrodinger operators in the Dirichlet case. Section 4 brie y reviews the multi-dimensional trace formulas in [25] and illustrates a possible abstract approach to some of these trace formulas in the special noninteracting case. In Section 5 we recall a dierent type of trace formula rst derived in [17] in dimensions one, two, and three based on point interactions. Section 6 nally describes a new uniqueness result for three-dimensional Schrodinger operators with spherically symmetric 2

2

2

2

2

2

2

2

2

1

potentials originally proven in [19].

2. Trace Formulas for Sturm-Liouville Operators. Let p; q; r 2 C 1(R) be real-valued, p; r > 0 and q bounded from below.

We then de ne the self-adjoint Sturm-Liouville operator in L (R; r dx) by hf = r1 [?(p f 0)0 + qf ]; (2.1) f 2 D(h) = fg 2 L (R; r dx)jg; g0 2 AC (R); hg 2 L (R; r dx)g; where AC ( ) denotes the set of locally absolutely continuous functions in  R. In addition, we de ne the Dirichlet Sturm-Liouville operator (2.2) hDy f = r1 [?(p f 0)0 + qf ]; f 2 D(hDy ) = fg 2 L (R; r dx)jg; g0 2 AC (Rnfyg); lim g(y  ) = 0; hDy g 2 L (R; r dx)g: # 2

2

2

2

2

2

2

loc

2

loc

2

2

2

2

2

loc

2

0

In order to derive trace formulas we will compare the resolvents of h and hDy . Let g(z; x; x0) and gyD (z; x; x0) denote the Green's functions (i.e., the integral kernels of the resolvents) of h and hDy respectively, g(z; x; x0) = (h ? z)? (x; x0); gyD (z; x; x0) = (hDy ? z)? (x; x0): (2.3) One veri es 0 (2.4) gyD (z; x; x0) = g(z; x; x0) ? g(z; x;g(yz;)gy;(z;y)y; x ) ; and hence d ln[g(z; x; x)]: Tr[(hDx ? z)? ? (h ? z)? ] = ? dz (2.5) To proceed further, we need a high-energy expansion, i.e., z ! 1, of the diagonal Green's function g(z; x; x). For that purpose we shall exploit the Liouville-Green transformation to nd a Schrodinger operator H which is unitarily equivalent to h and hence use known results for Schrodinger operators derived in [21], [22],[24]. De ne the change of variable 0 x t = t(x) = dx0 pr((xx0)) (2.6) x for an arbitrary but xed point x 2 R. Write P (t) = p(x(t)); Q(t) = q(x(t)); R(t) = r(x(t)) (2.7) and introduce the unitary operator 1

1

1

1

Z

0

0

U : L (R; r dx) ?! L (R; dt) (2.8) = (Uf )(t) = [P (t)R(t)] F (t); F (t) = f (x(t)); f 2 L (R; r dx): Theorem 2.1. ([10], see also [20]) The operator H = UhU ? in L (R; dt) explicitly reads Hf = ?f 00 + V f; (2.9) 0 f 2 D(H ) = fg 2 L (R; dt) j g; g 2 AC (R); Hg 2 L (R; dt)g; 2

2

2

1 2

2

1

2

loc

2

2

2

where

V (t) = RQ((tt)) + (R(t)1P (t)) 21 (R(t)P (t))(R(t)P (t))tt ? 41 ((R(t)P (t))t = qr((xx)) + 2pr((xx)) (r(x)p(x))xx + (r(2xr)(px()x))x pr((xx)) ? 4r(1x) (r(x)p(x))x 



2

2

2

!

2

3

2

4

x

:= v(x); x = x(t): Furthermore,

(2.10)

0 g(z; x; x0) = [r(xG)p((z;x)t(rx(x);0)tp(x(x))0)] = ; x; x0 2 R; z 2 C; 1 2

(2.11)

where G in the Green's function of H . Moreover,

HuD = UhDy U ? = ? dtd + V (2.12) with V given by (2.10), is the Schrodinger operator with a Dirichlet boundary condition imposed at the point u = xy dx[p(x)=r(x)]. Let GDu denote the Green's function of HuD . Then D (z; t(x); t(x0)) u gyD (z; x; x0) = [p(G (2.13) x)r(x)p(x0)r(x0)] = : Hence we nd, using known results for H [22], [24] that 2

1

2

R

0

1 2

1

Tr[e?hDx ? e?h]  # ]

X

0

`=0

s` (x) `;

Tr[(hDx ? z)? ? (h ? z)? ] jzj!1 1

1

]

(2.14) 1

X

z2CnC j =0

rj (x)z?j? ;

(2.15)

1

where C is a cone with apex at E := inf f(H )g and opening angle  > 0. Recursion relations for s` and rj are given by (cf. [22],[24]) (2.16) s` (x) = (?1)` r``(!x) ; ` 2 N ; r (x) = 12 ; r (x) = 21 v(x); (2.17) 0

+1

0

0

1

jX ?1

rj (x) = j j (x) ? j?` (x)r`(x); j = 2; 3; : : : ; ` 1 (2.18)

= 1; = 2 v; j j

j = ? 12 ` j ?` + 21 v ` j?` + 14 `;x j?`;x ? 21 `;xx j?` ; j = 1; 2; : : : : =1

0

1

X

+1

`=1

X

+1

Explictly, one computes





`=0

s = ? 12 ; s (x) = 12 v(x); etc. 0

1

(2.19)

The proof of (2.17) in [22] follows from the well-known dierential equation for ?(z; t) = G(z; t; t), namely ?2?tt(z; t)?(z; t) + ?t(z; t) + 4[V (t) ? z]?(z; t) = 1 (2.20) and the asymptotic expansion 1 i ? = ?(z; t) jzj!1 2 z ?j (t)z?j ; (2.21) j z2 nC 2

2

1 2

]

X

=0

C

with ?j (t) de ned in (2.18) but v(x) replaced by V (t). The next ingredient concerns the fact that g(z; x; x) is a Herglotz function for all x 2 R, i.e., g(
; x; x): C ! C is analytic, C = fz 2 C j Im z > 0g. Hence g allows a representation [3] g(z; x; x) = exp c(x) + d  ?1 z ? 1 +  (; x) ; (2.22) where (; x) is Krein's spectral shift function for the pair (hDx ; h) [28], satisfying 0  (; x)  1, (
; x) 2 L (R; d), and d(1 +  )? (; x) < 1. Although it will not be subsequently used, for completeness we show how to obtain an expression for c(x). Let z = i in (2.22). By taking realparts of (2.22) one infers that c(x) = Refln[g(i; x; x)]g: (2.23) Fatou's lemma permits the explicit representation (; x) = 1 lim arg[g( + i; x; x)] for a.e.  2 R (2.24) # and all x 2 R. We will normalize (; x) to be zero below the spectrum of h, i.e., (; x) = 0 for  < E . Using the spectral shift function, one can show that 1 dF 0()(; x) (2.25) Tr[F (hDx) ? F (h)] = +

+

+





Z



2

R

R

1 loc



2

R

1

0

0

Z

E0

whenever F 2 C (R); (1 +  )F j 2 L ((0; 1)); j = 1; 2 and F () = ( ? z)? ; z 2 C n [E ; 1). In particular, 1 (2.26) Tr[e?hDx ? e?h] = ? E de?(; x);  > 0; 2

2

( )

2

1

0

Z

0

Z1 D ? 1 ? 1 Tr[(hx ? z) ? (h ? z) ] = ? E0

; x) ; z 2 Cnf(hD) [ (h)g: d ((? x z)

(2.27) Combining (2.14) and (2.26) we obtain the general trace formula for Sturm-Liouville operators 2

2s (x) = v(x) = E + lim # 1

0

0

1 E0

Z

de?[1 ? 2(; x)]:

(2.28)

The Abelian regularization cannot be removed in general, see [18]. Higher-order trace formulas are given in the next theorem. Theorem 2.2. One infers `? E ` 1 de?t`? ? (; x) ; ` 2 N: + ` lim s (x) = ? 21 ; s`(x) = (?1) t# E `! 2 (2.29) 1

0

(

Z

0

1

0

0

h

1 2

i

)

>From the high-energy behavior of the Green's function we nd that pzg(z; x; x)]g? : p(x)r(x) = ifzlim [ #?1 1

(2.30)

In contrast to the Schrodinger case, the spectral shift function (; x) does not contain all the information necessary to construct both p and q in the Sturm-Liouville case, given the weight r. From (2.11) and (2.24) we see that in fact the spectral shift functions  and  of H and h respectively, are identical in the sense that (; x) = (; t(x)). For a given V we may construct (; t) associated with (HtD ; H ). By choosing any positive p 2 C 1(R) we may de ne the Sturm-Liouville operator h using (2.10) (or (2.11) for the Green's function). By construction, the pair (hDx ; h) will have (; x) as the corresponding spectral shift function. The behavior of (; x) is particularly simple in spectral gaps of h. Since p; q, and r are real-valued, g( + i0; x; x) is real-valued for  2 Rn(h). More precisely, suppose ( ;  )  Rn(h) and assume that (x) 2 ( ;  ) is an eigenvalue of hDx . Then one has 1

2

1

8
0. Then Floquet theory implies that

(h) =

1

[

n=1

[E

n?1); E2n?1 ];

2(

E < E  E < E 


0

1

2

3

(2.32)

(2.33)

and (2.34) (hDx ) = (h) [ fn (x)gn2 ; E n?  n (x)  E n; x 2 R; n 2 N: In the periodic case g( + i0; x; x) is purely imaginary on the spectrum, and hence N

2

1

2

0;  < E ; n (x) <  < E n; n 2 N : (2.35) (; x) = 1; E n? <  < n (x); n 2 N ; E n? <  < E n? ; n 2 N Combining (2.29) and (2.35) we obtain the following result. Theorem 2.3. Let p; q; r 2 C 1(R), p; r > 0 be periodic, p(x + a) = p(x), q(x + a) = q(x), r(x + a) = r(x) for some a > 0. Then 8 > >
> :1 2

2(?1)`+1 `!s

0

2

2

1

2(

2

1)

1

1 X ` [E2`n?1 + E2`n ? 2n (x)`]; ` (x) = E0 + n=1

` 2 N; x 2 R:

(2.36)

In particular, 2s (x) = v(x) = E + 1

0

1

X

n=1

[E n? + E n ? 2n (x)]: 2

1

2

(2.37)

Finally, we turn to the case where the Dirichlet boundary condition is replaced by a family of (Robin-type) self-adjoint boundary conditions. De ne h ;y f = r1 [?(p f 0)0 + qf ]; f 2 D(h ;y ) = fg 2 L (R; r dx) j g; g0 2 AC ([y; R]); R > 0; (2.38) lim [g0(y  ) + g(y  )] = 0; h ;y g 2 L (R; r dx)g: # 2

2

2

2

2

2

0

( = 0 corresponds to a Neumann boundary condition at y.) h ;y is unitarily equivalent (using the operator U in (2.8)) to the Schrodinger operator H ;u ;u = ? dtd + V; D(H ;u ;u ) = fg 2 L (R; dt) j g; g0 2 AC ([u; R]); R > 0; (2.39) 0(u  ) +  ( ; u)g (u  )] = 0; H lim [ g g 2 L (R; dt ) g ;  ;u ;u # 2

(

)

(

2

2

)

(

0

2

)

where V is given by (2.10), the boundary condition is located at y u(y) = dx pr((xx)) ; (2.40) x and  ( ; u) depends on u as well as on , viz., )x j = P ? (PR)t j : (2.41)  =  ( ; u) = pr ? (pr 2r x y R 2PR t u The Green's function of h ;y is given by (2.42) g ;y (z; x; x0) = (h ;y ? z)? (x; x0) 0 = g(z; x; x0) ? ( +(@ )+g(@z;)(x; y+)( @ +)g@(z;)gy;(z;y)y; x ) ; where we abbreviate (2.43) @ g(z; y; x0) = @xg(z; x; x0)jx y ; @ g(z; x; y) = @x0 g(z; x; x0)jx0 y ; etc. In this case ?( + @ )( + @ )g(z; y; y) is a Herglotz function such that Im[( + @ )( + @ )g( + i0; y; y)] < 0 for ? > 0 large enough. Krein's spectral shift function for the pair (h ;x; h) then reads  (; x) = 1 lim farg[( + @ )( + @ )g( + i; x; x)]g ? 1; 2 R; x 2 R;  2 R; # (2.44) and it satis es  (; x) = 0 for  <  ; (x) := inf f(h ;x)g; (2.45) Z

0









=

=

2

1

2

1

1

1

=

1

2

=

2

2

1

2

0

1

2

0

Tr[F (h ;x) ? F (h)] =

Z

1

 ;0 (x)

for functions F as in (2.25). In particular, we nd Tr[e?h ;x ? e?h]

dF 0() (; x)

1

X

 #] 0

`=0

s ;`(x) `;

(2.46) (2.47)

where

s ;`(x) = (?1)` r ;``(! x) ; ` 2 N ; 0

(2.48)

with (cf. [22],[24]), r ; (x) = ? 21 ; r ; (x) =  ( ; u(x)) ? 12 v(x); j? r ;j (x) = j ;j? (x) ? ;j?`? (x)r ;`(x); j = 2; 3; : : : ;

(2.49)

+1

0

2

1

1 X

1

1

`=1

;? = 1; ; =  ? 21 v; ; = 21  v + 21 vx ? 18 v + 81 vxx; 1 v + 3  v + 3 v (4v + v ) + 1 v (v ?  ) ? 1 v ? 1 v ;

; = ? 16 x 8 16 x 8 xx 8 xxx 64 xxxx j

;j = 18 2(v ?  ) ;`? ;j?`;xx ? (v ?  ) ;`? ;x ;j?`;x 1

2

0

3

2

2

2 2

Xh

+1

2

1

2

2

2

1

`=1

1

? 4 ;` ;j?` ? 4v(v ?  ) ;`? ;j?` ? 2vx ;`? ;j?`;x + ;`? ;j?` 2

+1

+1 8`

j

1

1

Xh =0

i

1

i

;`;x ;j?`;x ? 2 ;` ;j?`;xx ? 4( ? 2v) ;` ;j?` ; j = 2; 3; : : : : 2

(2.50)

Explicitly, one computes

s ; (x) = 21 ; s ; (x) =  ( ; u(x)) ? 12 v(x); etc. (2.51) The proof of (2.49) in [22] is based on the dierential equation for ? (z; t) = ( + @ )( + @ )G(z; t; t), namely 2[V (t) ?  ? z]?;tt(z; t)? (z; t) ? [V (t) ?  ? z]?;t(z; t) ? 2Vt(t)?;t (z; t)? (z; t) ?4f[V (t) ? z][V (t) ?  ? z] ? Vt (t)g? (z; t) = ?[V (t) ?  ? z] (2.52) and the asymptotic expansion 1 ?;j (t)z?j ; (2.53) ? (z; t) jzj!1 2i z? = j ? z2 nC 0

2

1

1

2

2

2

2

2

2

1 2

]

2

3

X

= 1

C

with ?;j (t) de ned as in (2.50) with replaced by  and v(x) by V (t). The analog of Theorem 2.2 now reads `  ; (x)` 1 s ;`(x) = (?`1) de?`? ? +  (; x) ; ` 2 N; + ` lim  #  ; x ! 2 (2.54) (

Z

0

1

0

h

) i

1 2

0( )

and, in particular,

s ; (x) =  ( ; u(x)) ? 21 v(x) (2.55) 1 ? ? +  (; x) : de = ? 12  ; (x) ? lim  #  ; x Our last example in this section will be the periodic case, assuming (2.32) to hold. 2

1

Z

0

0

h

0( )

1 2

i

In this case (h ;x) = (h) [ f ;n(x)gn2 ;  ; (x)  E ; E n?   ;n(x)  E n; x 2 R; n 2 N; with (h) given as in (2.33). The spectral shift function now reads 0;  <  ; (x); E n? <  <  ;n(x); n 2 N  (; x) = ?1;  ;n(x) <  < E n; n 2 N ? ; E n? <  < E n? ; n 2 N and the trace formula (2.54) in the periodic case now equals N0

0

0

2

8 > >
> :

1 2

2(

2(?1)` `!s ;`(x) = 2 ; (x)` ? E ` + 0

0

2

1)

1

0

(2.56)

(2.57)

1

[2 ;n(x)` ? E `n? ? E `n ]; ` 2 N; x 2 R: (2.58)

X

2

n=1

1

2

In the case ` = 1 we nd ? 2s ; (x) = v(x) = rq((xx)) + 2pr((xx)) (r(x)p(x))xx + (r(2xr)(px()x))x pr((xx)) ? (r(4xr)(px()x))x x 1 (2.59) =2 pr((xx)) ? (p(2xr)(rx()x))x + 2 ; (x) ? E + [2 ;n(x) ? E n? ? E n]: n Subtracting this equation from (2.37) yields 1 p ( x ) ( p ( x ) r ( x )) x ? r(x) ? 2r(x) = E ?  ; (x) + [E n? + E n ? n (x) ?  ;n(x)]: n (2.60) !

1

3

2

2

2

!2

4

X

0

2

2

0

1

2

=1

2

!2

X

0

2

2

0

1

2

=1

3. Matrix-Valued Schrodinger Operators.

In this section we extend the trace formula (2.28) to self-adjoint matrix-valued Schrodinger operators. General background on matrix-valued dierential expressions can be found, e.g., in [1], [40]. Unlike all other sections in this contribution, the material below is in a preliminary stage with more details appearing elsewhere. Let H in L (R)m  = L (R) Cm be a self-adjoint operator de ned by Hf = ?Imf 00 + Qf; (3.1) m 0 m f 2 D(H ) = fg 2 L (R) j gj ; gj 2 AC (R); 1  j  m; Hg 2 L (R) g; where f = (f ; : : :; fm)T , Im denotes the identity in Cm , and Q = (Qj;k ) j;km denotes a self-adjoint matrix satisfying Qj;k 2 C (R) bounded from below; 1  j; k  m: (3.2) Closely associated with the equation Hf = zf (3.3) is the rst-order 2m  2m system L(z)(f; f 0)T = 0; (3.4) where (f; f 0)T = (f ; : : : ; fm; f 0 ; : : : ; fm0 )T and d ? A(z); A(z) = 0 Im ; (3.5) L(z) = I m dx Q?z 0 2

2

2

2

loc

1

1

1

1

!

2

with I m the identity in C m. If (z; x) denotes a fundamental matrix for L(z), that is, L(z) (z) = 0; (3.6) or equivalently, 0(z; x) = A(z; x) (z; x); (3.7) ~ z; x) de ned by then ( ~ z; x) = (z; x)? ( (3.8) satis es the adjoint system ~ z; x)A(z; x): ~ 0(z; x) = ? ( (3.9) ~ z; x) are of the form Moreover, the fundamental matrices (z; x) and ( 0 z; x) (z; x) ; ( ~ z; x) = ~~(0 z; x) ?~~ (z; x) ; (3.10) (z; x) = 0 ((z; 0 x) (z; x) ? (z; x) (z; x) and one veri es that ? j00(z; x) + Q(x) j (z; x) = z j (z; x); ? ~j00(z; x) + ~j (z; x)Q(x) = z ~j (z; x); j = 1; 2: (3.11) In particular, assuming j (z), ~j (z) to be unique solutions of (3.11) (up to right resp. left multiplication of matrices constant with respect to x) satisfying (z;
) : = (z;
) 2 L ([R; 1))m; ~ (z;
) : = ~(z;
) 2 L ([R; 1))m; R 2 R; z 2 C n (H ); (3.12) the Green's matrix G(z; x; x0) of H becomes (z; x) ~?(z; x0); x  x0 (3.13) G(z; x; x0) = ? (z; x) ~ (z; x0); x  x0 and hence the resolvent of H is given by 2

2

1

!

!

1

2

1

2

1

1

2

1 2

2

1 2

8 < :

((H ? z)? f )(x) =

Z

+

+

dx0 G(z; x; x0)f (x0); f 2 L (R)m ; z 2 C n (H ): 2

1

Since

2

2

R

? j00(z; x) + j (z; x)Q(x) = z j (z; x); j = 1; 2;

(3.14) (3.15)

~j (z; x) are of the type ~j (z; x) = Aj; (z) (z; x) + Bj; (z) (z; x); j = 1; 2 (3.16) for matrices Aj;k (z), Bj;k (z), 1  j; k  2 in Cm constant with respect to x. Introducing the \Wronskian" W (; )(x) of m  m matrices  and by W (; )(x) = (x) 0(x) ? 0(x) (x); (3.17) one veri es that d W ((z); (z))(x) = 0 (3.18) dx 1

1

2

2

for solutions (z; x) and (z; x) of ? 00(z; x) + [Q(x) ? z] (z; x) = 0; ?00(z; x) + (z; x)[Q(x) ? z] = 0: (3.19) Relations (3.8), (3.12), (3.15), and (3.16) then yield ~(z; x) = W ( (z); (z))? (z; x) (3.20) and hence G(z; x; x) = ? (z; x)W ( ?(z); (z))? ?(z; x) = ?(z; x)W ( (z); ?(z))? (z; x): (3.21) The corresponding matrix-valued Dirichlet Schrodinger operator HyD in L (R)m then reads HyD f = ?Imf 00 + Qf; f 2 D(HyD ) = fg 2 L (R)m j gj 2 AC (R); gj0 2 AC (R n fyg); lim g (y  ) = 0; HyD g 2 L (R)m g (3.22) # j 1

+

1

+

1

+

+

2

2

loc

loc

2

0

and its Green's matrix GDy (z; x; x0), the analog of (2.4), is given by (3.23) GDy (z; x; x0) = G(z; x; x0) ? G(z; x; y)G(z; y; y)? G(z; y; x0): The analog of (2.5) then becomes Tr[(HxD ? z)? ? (H ? z)? ] = ?Tr[G(z;
; x)G(z; x; x)? G(z; x;
)] = ?Tr[G(z; x; x)? G(z; x;
)G(z;
; x)] = ?Tr m fG(z; x; x)? [ d G(z; x; x)]g dz d d = ? dz Tr m fln[G(z; x; x)]g = ? dz lnfdet m [G(z; x; x)]g; (3.24) where we used cyclicity of the trace, d G(z; x; x0) = m dx00 G(z; x; x00) G(z; x00; x0) ; (3.25) (H ? z)? (x; x0)j;k = dz j;` `;k j;k ` and Tr m [ln(M )] = ln[det m (M )] for matrices M in Cm . Moreover, Tr(
) and Tr m (
) in (3.24) denote the trace in L (R)m and Cm , respectively. Introducing the matrix-valued Green's kernel diagonal with respect to x (cf. (3.21)) ?(z; x) = G(z; x; x); (3.26) the matrix analog of (2.20) reads ??(z; x)?xx(z; x) ? ?xx(z; x)?(z; x) + ?x (z; x) + ?(z; x) Q(x) + Q(x)?(z; x) + 2?(z; x)Q(x)?(z; x) ? 4z?(z; x) = Im (3.27) and considerations along the lines of (2.20), (2.21) then yield 1 (3.28) ?(z; x) jzj!1 2i z? = ?j (x)z?j ; j z2 nC 1

1

1

1

1

C

C

X

2

=1

C

1

C

C

Z

R

C

2

2

2

2

1 2

]

with

X

=0

C

? (x) = Im; ? (x) = 12 Q(x); etc. 0

2

1

(3.29)

Similarly,

? dzd ln[G(z; x; x)] jzj!1

1

Rj (x)z?j? ; 1

(3.30)

R (x) = 21 Im; R (x) = 21 Q(x); etc.

(3.31)

]

X

z2CnC j =0

where

0

1

Next, de ne for all x 2 R the analog of (2.24) by (; x) = 1 lim Imfln[G( + i; x; x)]g for a.e.  2 R; # (; x) = 0;  < E := inf f(H )g; (3.32) where Im(M ), Re(M ), in obvious notation, abbreviate Im(M ) = 21i (M ? M ); Re(M ) = 12 (M + M ) (3.33) for matrices M in Cm . It follows from the results in [5] that 0  (; x)  Im for a.e.  2 R. (3.34) In the following denote by CR; the counter-clockwise oriented contour CR; = fz = E + ei j 32    2 g [ fz = E +  + i j 0    Rg [ fz = E + Rei j arctan(=R)    2 ? arctan(=R)g [ fz = E +  ? i j 0    Rg; R >  > 0: (3.35) Applying the residue theorem, taking into account that G(z; x; x), x 2 R, is analytic in z 2 C n (H ) and det[G(z; x; x)] 6= 0 for z 2 C n (H ) (cf. (3.23)), then yields 0 fln[G(z; x; x)]gj;k = 21i C dz0 fln[G(zz0 ;?x;zx)]gj;k R; 0 = 21i dz0 fln[G(z0; x; x)]gj;k 1 +z z0 0

0

0

0

0

0

I

I

CR;

2

1 ? z0 j;k 0 z ? z 1 + z0 CR; R = Refln[G(i; x; x)]gj;k + 1 d Imfln[G( + i0; x; x)]gj;k  ?1 z ? 1 +  E + o() + o(R? ) 1 1 ?  ; d ( ; x ) ?????! Re f ln[ G ( i; x; x )] g j;k j;k + R!1; # ?z 1+ E 1  j; k  m: (3.36) + 21i

I

"

dz0 fln[G(z0; x; x)]g

#

2

"

Z

#

2

0

1

"

Z

0

Thus

#

2

0

d ln[G(z; x; x)] = dz

Z

1

E0

d (; x)( ? z)? ; 2

(3.37)

and the matrix analog of (2.28) then reads Z

1

(3.38) Q(x) = E Im + z!limi1 E d z ( ? z)? [Im ? 2(; x)]; where we used a resolvent instead of a heat kernel regularization. De ning (3.39) (; x) = Tr m [(; x)]; one infers from (3.24) that 1 (3.40) Tr[(HxD ? z)? ? (H ? z)? ] = ? E d(; x)( ? z)? and that (; x) = 1 lim argfdet m [G( + i; x; x)]g; 0  (; x)  m for a.e.  2 R.(3.41) # Further details and applications of this formalism to inverse spectral problems will appear elsewhere. 2

0

2

0

C

Z

1

2

1

0

C

0

4. Multi-Dimensional Trace Formulas.

First, reporting on recent work in [25], we attempt to extend the leading behavior in (2.14), 2Tr[e?H ? e?HxD ] = 1 ? V (x) + o( ) as  # 0 (4.1) to arbitrary space dimensions  2 N. The key to such an extension is an appropriate combination of Dirichlet and Neumann boundary conditions on various hyperplanes through the point x 2 R taking into account that (4.1) is equivalent to Tr[e?HxN ? e?HxD ] = 1 ? V (x) + o( ) as  # 0; (4.2) where HxN = Hx denotes the operator (2.39) with a Neumann boundary condition at x 2 R. We start by introducing proper notations. In the following let V be a real-valued continuous function on R bounded from below and de ne the self-adjoint operator H = ?
u V (4.3) as a form sum in L (R ). Next, let A  f1; :::;  g and denote by jAj the number of elements of A. Moreover, let Bx ;   f1; :::;  g be the 2 blocks obtained by removing the hyperplanes Pj x = fy 2 R j yj = xj g from R , that is, Bx = fy 2 R j y` > x` if ` 2 ; y` < x` if ` 62 g and denote by P the power set of f1; :::;  g. The operator HA x is then L (Bx ) with Dirichlet boundary conditions on fPj x gj2A de ned to be ?
+ V on 2P and Neumann boundary conditions on fPj x gj62A. Theorem 4.1. [25] De ne C = A2B (?1)jAje?HA ;  > 0. Then the integral kernel of C  is given by  ?H (x; ?x0); x; x0 in the same orthant (4.4) C (x; x0) = 02; eotherwise. Moreover, C ;  > 0 is a trace class operator in L (R ) and Tr(C ) = 2  d xe?tH (x; ?x);  > 0: (4.5) 0

2

( )

( )

( )

L

;

2

( )

( )

P

;0

(

2

Z

R

( )

The proof of (4.4) in [25] is based on the method of images while the trace class property L (Bx ). of C and (4.5) follow from the direct sum decomposition of C in 2P Applying a Feynman-Kac-type analysis then yields the following  -dimensional generalization of (4.2). Theorem 4.2. [25] (4.6) Tr( (?1)jAj e?HA x ) = 1 ? V (x) + o( ) as  # 0: L

X

2

( )

;

A2P

While Theorem 4.2 represents a multidimensional trace formula for Schrodinger operators associated with unbounded regions in R , one can also prove new trace formulas for Schrodinger operators de ned in boxes. One obtains, e.g., Theorem 4.3. [25] Let V be continuous on [0; 1] . For A  f1; :::;  g, let HA be ?
+V on  L ([0; 1] ) with Dirichlet boundary conditions on the hyperplanes with xj = 0 or 1 and j 2 A and Neumann boundary conditions on the hyperplanes with xj = 0 or 1 and j 62 A. Let hV i be the average of V at the 2 corners of [0; 1] . Then (?1)jAjTr(e?HA ) = 1 ?  hV i + o( ) as  # 0: (4.7) 2

X

A2P

This result holds also for rectangular boxes j [aj ; bj ] but the rectangular symmetry is crucial in the proof of [25]. Similarly, one can prove Theorem 4.4. [25] Let V be continuous on [0; 1] . For A  f1; :::;  g let H~ A be ?
+ V on L ([0; 1] ) with Dirichlet boundary conditions on the hyperplanes with xj = 0 for j 2 A and Neumann boundary conditions on the hyperplanes with xj = 0 for j 62 A or xk = 1 for all k 2 f1; :::;  g. Then (?1)jAjTr(e? HA ) = 2? [1 ? V (0) + o( )] as  # 0: (4.8) =1

2

X

~

A2P

Finally, we mention an Abelianized version of a trace formula that Lax [29] derived formally in two dimensions. Theorem 4.5. [25] Let V be a continuous periodic function on R with V (x + n ; x + n ) = V (x ; x ) for all (x ; x ; n ; n ) 2 R  Z . Let HP ; HA ; HAP ; HPA ; HN ; and HD be the operators ?
+ V on L ([0; 1] ) with periodic, antiperiodic, AP; PA, Neumann, and Dirichlet boundary conditions respectively, where AP (resp. PA) means antiperiodic in the x (resp. x ) direction and periodic in the x (resp. x ) direction. Then Tr[e?HP + e?HA + e?HAP + e?HPA ? 2e?HN ? 2e?HD ] = ?1 + V (0) + o( ) as  # 0: (4.9) For a dierent kind of two-dimensional trace formula for V (x) comparing the heat kernels for H = ?
+ V and H = ?
with Dirichlet boundary conditions on a rectangular box, see [34]. Trace formulas for heat kernels of multi-dimensional Schrodinger operators in the short-range case have also recently been derived in [4]. Finally, we illustrate a possible new abstract approach to the trace formulas (4.7) based on certain commutation (supersymmetric) techniques in the noninteracting case where V (x) = 0; x 2 R . We need a bit of notation. Let H be a (complex separable) Hilbert space, F a closed densely de ned linear operator in H and de ne the self-adjoint operators H = F F ; H = FF  (4.10) 2

2

1

1

2

1

2 2

1

2 2

2

2

2

2

0

1

2

1

1

1

2

in H and

!

!

 (4.11) Q = F0 F0 ; P = 10 ?01 in H  H. Moreover, we denote by tr(:) the trace in H, by Tr(:) the trace in H  H, and by B(H) (resp. B (H)) the set of bounded (resp. trace class) operators in H. Lemma 4.6. One infers that (i) QP + PQ = 0: (4.12) (ii) (4.13) Q = H0 H0 : (iii) Fe?tH  e?tH F; F e?tH  e?tH F : (4.14) Proof. While (i) and (ii) are obvious, (iii) follows from Qe?tQ  e?tQ Q: Lemma 4.7. Assume B 2 B(H  H) is bounded and commutes with Q, i.e., QB  BQ. Suppose e?tQ , Q e?tQ 2 B (H  H), t > 0. Then d Tr[Pe?tQ B ] = 0: (4.15) dt Proof. d Tr[Pe?tQ B ] = ?Tr[PQ e?tQ B ] dt = Tr[PQe?tQ QB ] =


= Tr[PQ e?tQ B ] (4.16) using commutativity of Q and B and anticommutativity of Q and P in (4.12) and cyclicity of the trace. The fact that Q is unbounded is oset by the trace class hypotheses in Lemma 4.7. In fact, rewriting ?Tr[PQ e?tQ B ] = ?Tr[PQ(1 + jQj)? Q(1 + jQj)e?tQ B ] enables one to prove (4.16) in a trivial manner by reshuing Q(1 + jQj)? 2 B(H  H) as opposed to Q in (4.16). 1

!

1

2

2

1

2

2

2

2

2

2

1

2

1

2

2

2

2

2

2

2

2

1

2

2

1

Next we introduce the closed densely de ned linear operators Fn, 1  n   as in H and de ne H ;n = FnFn, H ;n = FnFn, 1  n   as in (4.10). Moreover, assume e?tHj;n ; Hj;n e?tHj;n 2 B (H); 1  n   and [Fm; Fn]  0; [Fm; Fn]  0; m 6= n implying [Hj;m; H`;n ]  0; j; ` = 1; 2; m 6= n: 1

2

1

We also denote

!

 (4.17) Qn = F0n F0n ; 1  n   in H  H as in (4.11) and de ne for any A 2 P (the power set of f1; 2; : : : ;  g) the self-adjoint operator HA = H ;n + H ;n: (4.18) X

0

X

1

n2A

2

n=2A

Then an abstract version of (4.7) in the noninteracting case reads as follows.

Theorem 4.8.

X

(?1)jAjtr(e?tHA ) =

X

0

A2P

A2P

(?1)jAj dimRan[PHA (f0g)]; 0

where PHA ( ),  R, denote the spectral projections of HA . Proof. One computes d jAj tr(H e?tHA ) jAj tr(e?tHA ) = ? ( ? 1) ( ? 1) A dt A2P A2P

(4.19)

0

0

X

X

0

=? by (4.15), where



X

n=1

0

(4.20)

0

Tr[PQne?tQn B;n ] = 0 2

2

!

B ; = 10 01 ;  b 0 ;n B;n = 0 b;n ; b;n = ? (e?tH ;m ? e?tH ;m ); m

(4.21)

11

!

Y

2

1

=1 =

m6 n

2

are bounded and commute with Qn. Thus the left-hand-side in (4.19) is independent of t and taking t " 1 then determines the right-hand-side of (4.19). Identifying An = 1


1



@ @xn D 1


1

in L ([0; 1]) ) with 2

@ = d (4.22) @xn D dx C1 ; ; 1  n   in L ([0; 1]) then yields (4.7) in the case V (x) = 0 since only the zero-energy eigenvalue of the Neumann operator H contributes on the right-hand-side of (4.19). More generally, if An has the tensor product structure An = 1


1 a n 1


1 in H = H


H , then clearly [Am; An]  0 and one evaluates



0 ((0 1))

2

0

1

X

A2P

(?1)jAj tr(e?tHA ) =

In the special case (4.22), where an =

0





Y

n=1

tr(e?tanan ? e?tanan ):

@ @xn D , one con rms that   tr e?tanan ? e?tanan = 1; 1  n  :

(4.23)

5. Trace Formulas and Point Interactions in Dimensions One, Two, and Three.

In this section we describe a dierent kind of multi-dimensional trace formula based on point interactions [2] and hence rank-one perturbations of resolvents rst derived in [17] in a slightly dierent form. Since point interactions (also called contact interactions or -interactions) are limited to  = 1; 2; 3 space dimensions, so will be our approach below. Assuming V to be real-valued, continuous and bounded from below on R ;  = 1; 2; 3, we introduce H = ?
u V as in (4.3). The resolvent of the self-adjoint Hamiltonian H;x, modeling H plus a point interaction centered at x 2 R (whose strength is parameterized in terms of  2 R), is de ned as follows (see, e.g., [2], [42]) (H;x ? z)? = (H ? z)? + D;x(z)? (G(z; x; :); :)G(z; :; x); z 2 Cnf(H;x) [ (H )g; (5.1) where ? ? ? (z; x);  = 1;  2 R [ f1g;  6= 0 (5.2) D;x(z) = ?  ? ? (z; x);  = 2; 3;  2 R; 1

1

1

(

1

? (z; x) = G(z; x; x); ? (z; x) = jlim [G(z; x; x + ) ? (2)? ln(jj)]; j# 1

1

2

0

? (z; x) = jlim [G(z; x; x + ) ? (4 jj)? ]; j#

(5.3)

1

3

0

and G(z; x; x0) denotes the Green's function of H . In analogy to (2.5) one then computes d ln[D (z)]: (5.4) Tr[(H;x ? z)? ? (H ? z)? ] = ? dz ;x Krein's spectral shift function for the pair (H;x; H ) is then introduced via 1 (5.5) d (;x?(z)) ; Tr[(H;x ? z)? ? (H ? z)? ] = ? E;x; with E;x; = inf f(H;x) [ (H )g and the normalization (5.6) ;x() = 0;  < E;x; : ;x() is related to D;x(z) as (; x) is to g(z; x; x) in (2.24). The high-energy expansion (see, e.g., [35], [41]) 1=4(?z) = + o(z? = );  = 1  =2 lim [G(z; x; x + ) ? G (z; x; x + )] = ?V (x) ?1=4z + o(z? ); jj# = ? = 1=8(?z) + o(z );  = 3 (5.7) then yields ?? ? 2i z? = ? 4i V (x)z? = + o(z? = ); =1 D;x(z) = (2)? ln((?iz) = ) + ~ ? (4)? V (x)z? + o(z? );  = 2 ; (5.8) ?i(4)? z = +  + i(8)? V (x)z? = + o(z? = );  = 3 where ~ =  + (2)? ? (2)? ln(2) with = :5772 : : : being Euler's constant. A combination of (5.4), (5.5), and (5.8) then implies the following trace formula. 1

1

Z

1

1

2

0

0

0

8 >
:

0

8 > > > < > > > :

1

1 2

1

1 2

3 2

1 2

1

1

1

1 2

3 2

1

1 1 2

3 2

1 2

1

1

1 2

Theorem 5.1. [17]  = 1:

V (x) =  = 2:  = 3:

8 > < > :

n

o

 = 1; limz#?1 ?z ? 2 1  H dz ( ? z)? 1;x() ; 1  + lim 2 i 8 i 1 z#?1 ? z + z = + ? z = E;x; d( ? z )? ;x () ; 6 3 3 3  2 Rnf0g: (5.9) R



inf[ (

2

)]

2

2

1 2

1 5 2



R

2

0

(

Z 1 ? 1 1=2 V (x) = zlim ?z + 4[(2) ln(?iz ) + ~ ] E dz2( ? z)?2;x() #?1 ;x;0 (

V (x) = 16  + zlim ?z + 4iz = + 2 #?1 2

2

1

Z

1 2

E;x;0

)

: (5.10)

)

dz ( ? z)? ;x() : (5.11) 2

2

Using the systematic high-energy expansion of limjj# [G(z; x; x + ) ? G (z; x; x + )] in terms of (multi-dimensional) KdV invariants (see, e.g., [35], [41]) one can extend Theorem 5.1 to higher-order trace relations in analogy to (2.29) and (2.54). In the special case where V  0, one obtains explicitly, ?? ? (?4z)? = ; =1 (5.12) D (z) = ~ + (2)? ln((?z) = );  = 2 ? = ~ + (4) (?z) ;  = 3; and 1 d (?(z)) : Tr[(H;x ? z)? ? (H ? z)? ] = ? (5.13) E; Here, for  = 1, 0;  < ? =4 0;  < 0 0; 0 1 + a();  > 0 (5.14) a();  > 0  0 ; ? ;  > 0 A();  > 0 (5.16) 0 writing A() = ?? arctan( = =4 jj), and ?(4) ;  < 0 :  =4;  < 0 ?e?  E; = ? 0;   0 0;  2 [0; 1] (5.17) =1  =2 =3 (0)

0

(0)

8 >
:

1 2

1

1 2

Z

(0)

1

(0)

(0)

1

2

(0) 0

8 >
:

1

1 2

8 > > >
> > :

4 ~

1

8 >
:

(0) 0

(

1 2

2

n

(

4 ~

2

6. A Uniqueness Result for Three-Dimensional Schrodinger Operators.

Finally, we brie y sketch a uniqueness result in the context of three-dimensional Schrodinger operators with spherically symmetric potentials originally derived in [19]. Consider the potential V : R ! R, V (x) = v(jxj); v 2 L ([0; R)]) for all R > 0 (6.1) and de ne the self-adjoint Schrodinger operator H in L (R ) associated with the dierential expression ?
+ v(jxj) by decomposition with respect to angular momenta. This represents H as an in nite direct sum of half-line operators in L ((0; 1); r dr) associated with dierential expressions of the type d ? 2 d + `(` + 1) + v(r); r = jxj > 0; ` 2 N : ^` = ? dr (6.2) r dr r A simple unitary transformation (see, e.g., [36], Appendix to Sect. X.1) reduces (6.2) to d + `(` + 1) + v(r) (6.3) ` = ? dr r and associated Hilbert space L ((0; 1); dr). Next, let G(z; x; x0), x 6= x0 denote the Green's function of H and de ne H; in L (R ),  2 R as in (5.1) (with x = 0) and the corresponding Krein spectral shift function ; () as in (5.5), i.e., ; () = lim ? Imfln[D; ( + i)]g a.e. (6.4) # 3

1

2

3

2

2

2

2

0

2

2

2

2

2

2

0

3

0

1

0

0

0

Then the following uniqueness result holds. Theorem 6.1. [19] De ne Hj ; Hj;j ; , j 2 R associated with ?
+vj (jxj), x 2 R , j = 1; 2 as above and introduce Krein's spectral shift function j;j ; () for the pair (Hj;j ; ; Hj ), j = 1; 2. Then the following are equivalent: (i)  ; ; () =  ; ; () for a.e.  2 R: (6.5) (ii)  = and V (x) = V (x) for a.e. x 2 R : (6.6) The proof of this result in [19] is based on detailed Weyl-m-function investigations associated with the angular momentum channel ` = 0. 3

0

0

1

1

2

1 0

2

0

2 0

1

2

3

Acknowledgments. We are indebted to B. Simon and Z. Zhao for joint collaborations which led to most of the results presented in this contribution and to B. M. Levitan for discussions which inspired Section 3. We are also grateful to G. Stolz for discussions on matrix-valued dierential expressions. F.G. would like to thank A. Friedman and the Institute for Mathematics and its Applications, IMA, University of Minnesota, USA and the Department of Mathematical Sciences, NTH, University of Trondheim, Norway for the great hospitality extended to him during a month long stay at each institution in the summer of 1995. F.G. also thanks B. Simon and D. Truhlar for their kind invitation to the IMA workshop \Multiparticle Quantum Scattering with Applications to Nuclear, Atomic and Molecular Physics". Moreover, we are both indebted to T. Homann-Ostenhof for the kind invitation to the Erwin Schrodinger International Institute for Mathematical Physics,

ESI, Vienna, Austria for a period in June and July of 1995. Financial support by the IMA, the Norwegian Research Council, and the ESI is gratefully acknowledged.

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Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

E-mail address : [email protected]

Department of Mathematical Sciences, Norwegian Institute of Technology, University of Trondheim, N{7034 Trondheim, Norway 2

E-mail address : [email protected]