Aug 22, 1995 - We consider a class of nonselfadjoint quadratic operator pencils generated by the equation, which governs the vibrations of a string with ...
Integr Equat Oper Th Vol. 25 (1996)
0378-620X/96/030289-4051.50+0.20/0 (c) Birkhauser Verlag, Basel
B A S I S P R O P E R T Y O F E I G E N F U N C T I O N S OF N O N S E L F A D J O I N T OPERATOR PENCILS GENERATED BY THE EQUATION OF NONHOMOGENEOUS DAMPED STRING Marianna A. Shubov We consider a class of nonselfadjoint quadratic operator pencils generated by the equation, which governs the vibrations of a string with nonconstant bounded density subject to viscous damping with a nonconstant damping coefficient. These pencils depend on a complex parameter h, which enters the boundary conditions. Depending on the values of h, the eigenvalues of the above pencils may describe the resonances in the scattering of elastic waves on an infinite string or the eigenmodes of a finite string. We obtain the 7asymptotic representations for these eigenvalues. Assuming that the proper multiplicity of each eigenvalue is equal to one, we prove that the eigenfunctions of these pencils form Riesz bases in the weighted L2-space, whose weight function is exactly the density of the string. The general case of multiple eigenvalues will be treated in another paper, based on the results of the present work.
1.
Introduction. In the present paper we develop the spectral analysis for a one-parameter family
of nonselfadjoint quadratic operator pencils. These pencils appear in the study of the wave equation which describes a nonhomogeneous damped string. The main result of the work is the proof of the fact that the systems of the eigenvectors and associated vectors of the above mentioned pencils form unconditional bases in the corresponding Hilbert space. There exists an extensive literature on operator pencils (it can be traced, for example, through the references in the book [1]). However, known abstract results cannot be applied to the pencils we consider in this work. In the present paper we prove the above result in the case when there are no associated vectors, i.e. the proper multiplicity (see [1, 2]) of each eigenvalue is equal to one. The proof of the main result in the general case will be given in another paper. This proof will be based on the results of the present work. We have considered the case of
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simple eigenvalues separately, to make the proofs more transparent. In the proof of the basis property of the eigenfunctions we will use the asymptotic formulas for the eigenvatues and the eigenfunctions. These formulas are, also, derived in the present work and might be of interest in themselves. We mention that in this work we consider the string with a nonsingular density function. Similar results are correct for the densities which have singularities corresponding to an infinite condensation or an infinite rarefaction of the matter. However, the proof of these results for singular densities is significantly more complicated and requires absolutely different techniques. The study of singular cases has begun in our paper [3] (see, also, [4, 25]) and will be completed in our two forthcoming works. (On the way to the proof of the basis property in the latter three works we obtain several results which might be of interest in themselves (see[3] for details)). The study initiated in the present paper may have interesting continuations in different directions, including applications to control of damped distributed parameter systems, but, nevertheless, this paper is self contained: it has the proofs of all statements formulated in Section 2. The precise formulations of the results of the present work will be given in Section 2. Here we describe the physical problem which leads to the above mentioned pencils. As it is well known, the vibrations of a nonhomogeneous damped string axe governed by the wave equation
u . - fi_L_~ p(x) ~
+ 2d(x)Ut = O,
(1.1)
where p(x) > 0 is the density of the string and d(x) > 0 is the damping. In our study we assume that x E [0, r
i.e. the string is semi-infinite, and that U(0, t) = 0 for any t e [0, oo).
(1.2)
In addition to the boundary condition (1.2) we introduce below a one-parameter family of conditions at the point x = a (a > 0). Depending on the values of the parameter these conditions describe different physical problems, which we mention below. Both the assumption that the string is semi-infinite and the condition (1.2) are not crucial for our analysis. All our methods can be extended in a straightforward manner both to the case of infinite string (x E R) and to the case of some other boundary conditions at x = 0. Our main reason to choose the condition (1.2) is the connection between Eq. (1.1) and the radial wave equation describing the propagation of acoustical waves in a spherically symmetric nonhomogeneous medium with energy dissipation. We study the latter equation in our forthcoming work.
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291
Our assumptions about the density p(x) and damping d(x) are the following: p(z)=l
and d ( z ) = 0
for x E [ a , oo),
(1.3)
where a is a fixed number. (1.3) means that both the inhomogeneity and the damping are concentrated on a finite interval [0, a). We, also, assume that the density p is a smooth and positive function on a finite interval [0, a). It turns out that the behavior of p at the end z = a is crucial for us. In this paper we consider densities p which are either continuous functions at x = a or with a finite jump at x = a. The cases of singular behavior of p, when p(a - O) = oo (infinite condensation of the matter) or p(a - O) = 0 (infinite rarefaction of
the matter) axe considered in another paper (see [3]). To formulate our problem and to describe the above mentioned condition at x = a we have to come back to Eq. (1.1). We look for a solution in the form
t) =
(1.4)
where A is a complex parameter. Substituting (1.4) into (1.1) we see that u must satisfy the equation ux~ + A2p(x)u - 2 i A d ( x ) p ( x ) u = 0.
(1.5)
Let us add the following boundary conditions
u(0) = 0,
(1.6)
(u~ + iAhu)(a) = 0,
(1.7)
where h is an arbitrary fixed complex number. (1.6) follows from (1.2); (1.7) will be briefly discussed below. Eq. (1.5) and the conditions (1.6) (1.7) define a nonselfadjoint quadratic operator pencil, which will be our main object of interest. DEFINITION
1.1 We say that A E C is an eigenvalue if the problem (1.5) -
(1.7) has a nontrivial solution. This solution is called an eigenmode or an eigenfunction.
Let us mention some particular cases of the problem (1.5) - (1.7). In the case h = 0 we have the Neumann boundary condition at x = a and our problem describes, in fact, the vibrations of a finite string with the left end fixed and the fight end free. In the case when h ~ oo the condition (1.7) turns into the Dirichlet boundary condition u(a) = 0. So, in this case we deal with the vibrations of a finite string with both ends fixed. In the case when h = 1, (1.7) turns into the so-called Zommerfeld radiation boundary condition. In this case the above defined eigenvalues are called resonances and our problem describes the resonance
292
Shubov
phenomena in the scattering of acoustical waves on the semi-infinite string. The physical origin of the resonance problem is discussed in a more detailed manner in the author's work [4], where the propagation of acoustical waves in a nonhomogeneous spherically symmetric medium (without dissipation) is considered (see, also, [5, riD. As it was already mentioned, we consider the condition (1.6) at x = 0 only for the simplification of the exposition. All our proofs can be extended in a straightforward manner to the case when at x = 0 we have the condition (u~: + iXku)(O) = 0, where k is a fixed complex number. Now we give a brief description of the organization of the paper. In this work we obtain three main results. In Section 2 we formulate these results and give all necessary definitions and formulations of auxiliary propositions. In Section 3 we obtain the first main result. We show that the problem (1.5) - (1.7) defines a countable set of eigenvalues {),,, n E Z\{0} = Z'}, which is symmetric with respect to the imaginary axis for h real and derive an explicit asymptotic formula for these eigenvalues. To do this we obtain asymptotic representations for the solutions of Eq. (1.5) and, in particular, for the eigenfunctions {Fn(z), n E Z'}. It follows from our results that (Ira X~) (Re h) > O, I m X~ ~ C (C is a constant such that C Re h > 0) and Re X~ --* +or as n --* ~=co, where the constant C can be expressed explicitly in terms of p(x) and d(x). In the case h = 1 the corresponding solutions U=(x, t) = eZiX=tu,(x) of the wave equation (1.1) describe long-lived oscillations of the string whose amplitudes decrease exponentially. The quantity r= = (Ira)~n)-i is called the life-time of the resonance, and IRe X, I is called the frequency. So, we have a high frequency (]Re X,~]--* ~ ) series of resonances, whose life-times r, ~ C -1. The above mentioned asymptotic formulas for the eigenvalues are obtained based on the fact that the eigenvalues coincide with the roots of a certain entire function, which we call the generalized Jost function. In Section 4 we obtain our second main result. Assuming that all roots of the generalized Jost function are simple, we show that the set of eigenfunctions {F,~(z)},,>I is complete in the space L~(0, a) for every real h and for h = co, which corresponds to the string with fixed ends. In the proof of this result we use the results obtained in our paper [4]. At last in Section 5 we prove our third main result. Under the same assumptions as in Section 4 we show that the above set of eigenfunctions is, in fact, a Riesz basis in the space L~(0, a). To obtain this result we reduce the problem to the question about the Pdesz-basis property of the system of the exponentials {e ~"~} (see [5]- [101). The extension of all these results to the case of the generalized Jost function with multiple roots (which is equivalent to the existence of associated vectors) and to arbitrary h E C will be given in our next paper. Among the main results of the present work we can mention an important state-
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293
ment contained in Remark 4.2 c). Namely, the system of the eigenvectors and associated vectors of the operator ~:h defined by (4.2) - (4.5) below forms a Riesz basis in the energy space ~ defined in (4.1).
2.
Statement of main results. In this section we formulate our main results on the asymptotics of the spectrum
of the problem defined by (1.5) - (1.7) and on completeness and the basis property of the set of eigenmodes. We begin with the precise formulation of the properties of the functions p and d. About the density we assume, that p(~) = 1 for ~ > a; p ~ C2[0, a); p(x) > 0 for ~ ~ [0, a] and p(~ - 0) > 0.
(2.1)
In the case when h is real we need an additional restriction
p(a - 0) # Ih].
(2.2)
About the damping we assume that d E W ~ ( 0 , a),
d(z)>_Oforxq[O,a),
d(z)=0forz>a.
(2.3)
(Here W~ is the Sobolev space of functions whose first weak derivatives are integrable.) Notice, that with our conditions on p and d we have:
3,t= X =
dt > O, 34 < ~ , d(t)
> 0,
.~" < ~o.
(2.4) (2.5)
These quantities will be important in the following. We mention that, in fact, the asymptotic behavior of eigenvalues depends on the behavior of the density function p(z) only at the vicinity of the endpoint z = a. So, we can admit singularities and zeros of p somewhere in the inner points of the interval [0, a), even at the endpoint z = 0 (see [4] for the case of d = 0). The leading term of the asymptotics will be the same. We do not allow this behavior only for the sake of the simplification of the asymptotic analysis. If we allowed singularities or zeros of p in [0,a], then (2.4) and (2.5) would be additional restrictions on p. Our first main result - the asymptotics of the eigenvalues and the eigenfunctions of the problem (1.5) - (1.7) - is given by Theorem 2.2 below. It is convenient for us to begin with the statement of an auxiliary result-Theorem 2.1. This theorem contains the information which will be used both in the formulations and in the proofs of all main results of this work.
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2.1 a). For any A E C there exists a unique solution l ( A , z ) of
THEOREM
Eq. (1.5) which satisfies the condition (1.7) and the normalization condition { h -a,
ifh#O,
1,
if h = O.
J(A,a) =
(2.6)
We will call the solution I(A, x) the generalized lost solution. (Recall that J(A, x) is the well known lost solution if h = 1 (see [11])). For each z E [0, a] the generalized lost solution is an entire function of A. b). We call the function
J(A) = J(A, O)
(2.7)
the generalized ,lost function (in the case h = 1 it is known as the lost function (see [11])). For each h the generalized lost function is an entire function of the first order and of the
exponential type M (see (e.#). c). The generalized lost function J(A) has an infinite set of roots. All roots, except for, maybe, a finite number of them are simple. The simple roots are separated: inf
[A. - A,,,I > O.
(2.S)
For real h the set of roots is symmetric with respect to the imaginary axis and the multiplicities of any two symmetric roots are equal. Only a finite number of roots may be purely imaginary. d). The generalized lost function is a sine-type function (see Definition 2.1 below). DEFINITION
2.1 An entire function ~(A) of exponential type is said to be a
sine-type function (see [9, 1~]) if the following two conditions are satisfied: a) all roots of ~o(A) are separated and are located in a strip parallel to the real azis; b) for each line e parallel to the real axis and lying outside the above strip there exist two constants C2(g) > Ca(g) > 0 such that on this line:
ca(e) _< I~(m)l _< c=(e). REMARK
2.1 a) Due to the symmetry of the set of roots in the case of real
h, it is convenient to use both positive and negative integers for the numeration of the roots. As it will be clear from our asymptotic formulas, such numeration is, also, natural for complex h. More precisely, we represent the set of all roots of the lost function in the form
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295
{A,, n E Z' = 7.\{0}}. A_, = - X , for Re A, ~ 0. This numeration of roots requires an additional explanation since some of the roots may be purely imaginary. This explanation will be given after the formulation of Theorem 2.2. In Theorem 2.2 we use the asymptotical numeration of the roots. The precise absolute numeration will be important for the problems of completeness and the basis property of the eigenmodes. b) It is obvious from the definition of the Jost function that its roots {A,},~z, coincide with the eigenvalues of the problem (1.5) - (1.7), and, therefore, the functions F,,(x) = J(A,,, x) are exactly the eigenmodes. c) We emphasize that the statement d) of Theorem 2.1 contains , in particular, the following information: the set of all roots belongs to a strip parallel to the real axis and these roots do not have points of accumulation except ~ . d) The statement d) of Theorem 2.1 will play a crucial role in the proof of the Riesz-basis property of the set of eigenmodes (Theorem 2.5). Now we are in a position to formulate the first main result of this work. THEOREM
2.2 Let
a+ = h -1 (p(a)) -~ 5=(p(a)) 88, Co = 1/2vf~-~a-,
(2.9)
then the following asymptotic formula for the eigenvalues of the problem (1.5)- (1.7) holds for any h E C,h ~ 0
= Am + O(Inl-1), Inl - ~ ~ , where .,i,,= ~-' (n + l/2sgn n)~" + i~-' [1/21n (,+o,:') +.,V'], ,~,,
(2.10)
where .M and Af are given by (2.4), (2.5), and under In we understand the princ@al value of the logarithm. The following asymptotic representation for the quasimodes holds for any hEC, h~O --1/4
a
a
i/21n( l + 1 for the numeration of the roots with positive real parts and the integers n < - I - 1 for the numeration of the roots with negative real parts. The whole set of roots Am can be ordered in such a way so that Re A,, < Re A,+l. If several roots have the same real part then their order is immaterial. All this explains Remark 2.1 a): we see that the whole set of roots can be written as {A,,, n E Z'}. In the case when h is real, A_, = - X , if Re A,, # O. If the roots are not simple then we can represent the whole set in the same form if the roots are counted with their multiplicities. However, as it was already mentioned, the case of multiple roots will be treated in our next paper. Our second and third main results deals with the completeness and the basis property of the set of eigenmodes. Before we state these results we formulate several definitions and auxiliary propositions. As it can be seen from Theorem 2.2, the set of quasimodes is approximated by the set of cosine functions with comphcated arguments depending upon A,~. It turns out, that to prove the basis property of the set of functions whose asymptotics is given by (2.11)
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297
(or by (2.13)) we should start from the system of exponentials. More precisely, we need to have the answer to the question about the basis property of the system of exponentials:
The latter question (as it will be shown in Section 5) can be reduced to the question about the basis property of the standard system of nonharmonic exponentials {eiX~}neZ, in the Hilbert space L2(I), where I is an arbitrary interval of the real axis of the length 2.A4, where M is given by (2.4). This reduced problem is very well known (see [8,9]) and can be formulated as follows: when can a given functionf E L2(I) (where [ is a finite interval of the real axis R) be expanded in a Dirichlet series with respect to the exponentials with complex frequencies: f = ~
ane '~"~,
(Z' = Z\{O}).
(2.14)
nEZ'
In general, the family of exponentials {ei~"~)~ez, is not orthogonal in the Hilbert space L2(I) and even more, it is not necessarily complete. So, we distinguish two problems: completeness of exponentials {eia~}neZ, in L2(I) and (in the case of completeness) the possibility of expansions with respect to this system. As it is well known, the convergence of expansions with respect to any complete orthonormal system { ~ } in L2(I) is unconditional, i.e. the corresponding Fourier series converges to the same sum after any permutations of its terms. The latter fact becomes true for any system {~,~} obtained from an orthonormal basis by means of any bounded and boundedly invertible transformation of L2(I). Now we introduce several definitions. Since these definitions will be applied to our problem, it is convenient to assume that the index n runs over the set Z ~= Z\{O}. D E F I N I T I O N 2.2 For any positive functions qax(x) and qa~(x) defined on the interval I C R or for two sequences a,~ > 0 and bn > 0 (n E Z') we will write (2.15) if there exist two positive constants Cx and C2 such that for all x E I or n E Z'
Cl~2(x) __~~l(X) ~ C2~2(2~) or D E F I N I T I O N 2.3 The system {r
Cla n ~ bn ~ C2an.
(2.16)
from L2(I) is said to be almost nor-
realized if ]lenll • 1.
(2.17)
It is clear from the Definition 2.2 that (2.17) is equivalent to the conditions inf Ilenll > 0,
n~Z'
sup I1r
nEZ'
< O0.
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D E F I N I T I O N 2.4 Any complete almost normalized system {r in L2(I) is called a Riesz basis (R-basis) if there exist an orthonormal basis {~o,~),,ez, and bounded, boundedly invertible operator A such that qo, = Ar The operator A is called an orthogo-
nalizer of the system {r D E F I N I T I O N 2.5 A family of nonzero vectors {r in a Hilbert space H is called an unconditional basis in H if a) the family {r spans the space H (i.e. the set of all finite linear combinations of these vectors is dense in H); b) there exist two positive constants Cx and C2 such that for every finite set of complex numbers {am} the following inequalities hold:
ca ~ la~l=l[r n
2 ___ II~ a~r n
= < C2 ~ la~l=llr ~.
(2.18)
n
This definition requires some explanation. According to the standard definition of an unconditional basis {r
(in any Hilbert space) every element f of the space can
be uniquely decomposed in an unconditionally convergent series f = ~ e z ' a~r
The latter
definition is equivalent to Definition 2.5 due to the following property of unconditional bases (see [2, 7, 14]): a complete system {f,}~ez, forms an unconditional basis if and only if the "approximate Parceval identity" (2.14) holds. As it follows from Definitions 2.4 and 2.5, every R-basis is unconditional and every unconditional almost normalized basis is an R-basis. If the system of exponentials {eix"~},~ez, forms an unconditional basis, it may happen that lira A,[ ~ oo as In[ ~ oo. This cannot happen if this system is an R-basis. It is well-known that the problem of R-basis property of exponentials is closely related, to the theory of entire functions (see [10, 12]). Namely, an entire function is called a generating function for the family {elX-~}.~z, if its zeros coincide with the points A,~. In our study of the R-basis property in Section 4 we will use the following well-known result. T H E O R E M 2.3 (13.Ya. Levin, V.D. Golovin [15]). Assume that the set of complex points {A~) is separated (see (2.8)) and there exists a generating function of the family of exponentials {eia"~}~z, which is a sine-type function with the width of the indicator diagram (see [12, 16]) equal to a, a > O. Then {e'~'~}~z, forrr~ an R-basis in L2(I) where I is any interval of the length a. Now we are in a position to formulate our results on completeness and R-basis property of the eigenmodes. Let us split the whole set of the eigenmodes into two subsets ~+ = {F~(x)},,>o, ~- = (F~(x)}, O, 0 < k < m , - 1 } U ~ + ,
(2.24)
and similarly 5 - with n < 0 and ~_ instead of ~+. With ~+ and 5 - defined by (2.21) and (2.22) Theorems 2.4 and 2.5 remain true in the case of multiple roots of the Jost function. The proofs of these theorems will be given in our next paper. Now we give a short proof of the statement a) of Proposition 2.1. Let us multiply Eq. (1.5) by fi and integrate over the interval [0, a]. Using the boundary conditions (1.6), (1.7) we obtain -
II.'ll +
= - 2 i A i i v ~ u l [ 2 - iAhlu(a)l ~ = O.
(2.25)
In (2.25) and below we denote the norm in the space L~(0, a) by I[" H. Setting A = i#, # = 12 we have from (2.25) [[u'[[2 + ~211u112 - 2~llvfdu}[ 2 - #h[u(a)[ 2 = 0.
(2.26)
It follows from Eq. (2.26) that for real # one may have a nontrivial solution of Eq. (1.5) satisfying (1.7) only if h is real. Moreover, from Eq. (2.26) one can see, that for # = 0, due to (1.6), it follows that u = 0.
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Shubov
e) In the conclusion we mention that the present paper together with the forthcoming work [3] can be considered as a continuation of the author's work [4]. In [4] we studied resonances and resonance states in the scattering of acoustic waves by a spherically symmetric nonhomogeneous medium without damping. In that paper we dealt with the radial wave equation 1 V.
-
(
g(s -
r2
In contrast with (1.1), this equation does not contain the damping term
d(r)Ut but
contains
the centrifugal term l(~ + 1)r-2U. We expect that using the methods of [4] it is possible to extend the results of the present work and the above mentioned forthcoming work to the radial wave equation with damping Uu - p(~) U~
l ( / r2+ 1) V + 2d(r)Ut = 0.
This will allow us to treat resonances and quasimodes in the scattering of acoustical waves in a nonhomogeneous medium with dissipation of energy.
3.
Asymptotic representations for eigenvalues and quasimodes. This section is devoted to the proof of Theorems 2.1 and 2.2. We start from the
construction of two linearly independent solutions of Eq. (1.5). The first result of this section (Theorem 3.1) describes the asymptotical behavior of these solutions and their derivatives. The estimates obtained in Theorem 3.1 are considered as rough estimates and we use them for the next step: Theorem 3.2. In the latter theorem we sharpen these estimates and in Theorem 3.4 use these improved estimates to derive the asymptotic representation for the generalized Jost solution. Before we formulate and prove these theorems we have to do some preliminary work. First of all let us reduce Eq. (1.5) to a certain standard form. Let us introduce a new independent variable ~(x). We assume, that the inverse function of ~(x) is also exists. In what follows differentiation of any function with respect to ~ will be denoted by "prime." In terms of the new variable Eq. (1.5) obtains the form
u"-x'(x')-lu' + )~2p(~(x))(x')2u- 2i)~d(x(~))p(x(~))(x')2u = O,
(3.1)
To get rid of the first order derivative of the solution u we introduce a new function w : u = v ~ w . It is easy to show that w(~) satisfies the equation
WH--~ (0.Sg"(X') -1 -- 0.75(X•)2(X')-2)W "~ ~2p(x(~))2W-- 2,)ld(z(~))p(~g(~))21/J = 0. (3.2)
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Denoting by ~(~) and d(~) the followingfunctions
~(~) =
0.7~(~")~(~') -~ - o.s~"(~') -~,
~(~) = a(~(~)),
(3.3)
we arrive at the equation for w W tt
§
-
_-
(3.4)
Let us take ~(z) such that p(x)(x') 2 = 1. Integrating the latter equation we find a convenient change of variable
f0 It is convenient for us to use a new spectral parameter m instead of A : A = ira. In terms of and m Eq. (3.4) obtains the form
w" - mSw + 2md(~)w = ~ow.
(3.6)
Let us look for solutions of Eq. (3.6) when Re m > 0. (All the steps for Re m < 0 can be done in a similar way). Instead of Eq. (3.6) we consider two Volterra integral equations, which define uniquely two linearly independent solutions of Eq. (3.6). The first solution denoted by F(m, ~) decre~es when ~ --* ~ and Re m _> 0. F(m,~) satisfies the following Volterra integral equation
F(m,~) = e -'~('-~) - m - 1 / ~ 1 7sinh(m(~ 6 - r/)) [qo(r/) - 2md(y)]F(m,y)&h
(3.7)
where .s is given by (2.4). The second linearly independent solution behaves as cosh(m~) when ~ --* 0 and satisfies the following Volterra integral equation ~(m,~) = cosh(m~)+ m -1 f0' s i n h ( m ( ~ - r/))[qo(~/)- 2md(y)]ql(m, T1)dr1.
(3.8)
In both equations the upper limit of integration does not exceed .h,4 because ~o(~) = d(~) = 0 for ~ > A4. The following statement holds. T H E O R E M 3.1 If the density p(x) and the damping d(z) satisfy the conditions (#.1) and (#.e) then for each ~ > 0 both solutions F(m,~) and k~(m,~) are entire functions of m. The following estimates hold for Re m > 0
[F(m,~)e'~(e-~)[ < C, [f'(m,~)em(e-~")l < Viral,
[kO(m,~)e-'~el < C; I~'(m,~)e-'~e[ < Viral.
(3.9) (3.10)
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(In this theorem the conditions on p and d can be weakened. However, this is not very important for us. So we use here the conditions (2.1), (2.2) which will be necessary in the analysis below.) Proof. Notice that due to (3.5) the function qo from (3.3) can be written in the form
= _5(p~)2(4p)-i + p=~p-2.
(3.11)
From (3.11) we get qo E LI(0, a). Let us take for definiteness Eq. (3.7) and solve it by the method of successive approximations. For the n-th term of the successive approximations series we can write the following estimate:
I ol
1
f=,,~-2 ]e'(=r
], o~
[l l + I ll l]
. . . . ')1 []q0]+ ]2md]] [e. . . . .
[J l + i dt]
']dx,=-i < -'[e-'r -l-'l-'~mn']--~" ([\.,Z~ (]~] + ]2md]) dr/)" .
From the latter estimate the first approximation from (3.9) follows immediately. To obtain (3.10) we notice, that
F ' = - m e -'`(r
- f oo cosh (rn(~C- r/))[~p- 2md] F dr/.
(3.12)
Due to (3.9) we can easily estimate the integral term in (3.12) and obtain the first estimate (3.10). The same way we can obtain the estimates for kO and its derivative. Theorem is shown. In what follows we will use Theorem 3.1 only to improve the estimates for the solutions F and @. To do this we introduce more sharp change of variable. Let us return to Eq. (3.6). Since this equation does not have any singular points we can look for a solution of the corresponding homogeneous equation in the form of an expansion with respect to the complex parameter m in the neighborhood of infinity (see [13] Murray)
w = g(~)e"~(~) ~ f,~(~)rn-", fo(~) = 1.
(3.13)
r~=0
Substituting w in the form (3.13) into Eq. (3.6) and collecting all terms having m 2 we obtain the following equation for r
(r
= 1.
(3.14)
Collecting all terms having m we obtain following equation for g(~) g'(~) =-d(~)g(~).
(3.15)
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305
Based on (3.14)and (3.15) we can sec that the solutionof Eq. (3.6)with ~ = 0 asymptotically behaves as the function of the argument ( m ~ - f0~ d(r/)dr/). N o w let us introduce exactly this argument as our new variable. Let
Notice that r is complex because m is complex. N o w rewrite Eq. (3.6) in terms of r. Let w~ be the derivativeof w with respect to r and recallthat w' is the derivativewith respect to ~. Based on the relations w' = w~.(m - d(~)) and w" = w~.,(m - d(~))~ - w,~(~), Eq. (3.6) can be given in the form w,,-w=
- w, (m - d) 2
~ w+ - w. (m - d) 2 (m - d) 2
(3.17)
Notice that formally the denominator in (3.17) can be equal to zero, but, in fact, we are interested in the asymptotic behavior of the solutions when Iml --~ oc. So, we consider only m's such that Im - g(~)l > ~ > 0. REMARK
3.1 At this moment we introduce an important agreement, which
will be used in the rest of this section. Notice, that the complex parameter r defined by (3.16) is a function of the real parameter x e [0,a]. The parameter r ( x ) runs over a certain curve in the complex plane when x runs over the interval [0, a]. The endpoints of this curve are r(0) = 0 and r(a) = mA4 -A/', where .A4 and N" are real constants defined by (2.4), (2.5). In the following we will use the symbols over the segments of the
9dr/and 9 dr/to denote the integration /0" above described curve, which have the endpoints either 0 and r or
r and m 3 d - A/" respectively. So, in the above integrals the integration paths depend on m. However, we do not indicate this dependence explicitly. Let ~'(m, r) be the decreeing solution of Eq. (3.17), satisfying the following Volterra integral equation
) (m-g) P,
-v__ p] J
(3.18)
Let ~ ( m , r ) be the solution of Eq. (3.17) that behaves as c o s h r when r --+ 0. It satisfies the following Volterra integral equation r
r) = cosh r + f0 sinh(r - r/)
One can see that two equations (3.7) and (3.18) define one and the same solution of Eq. (3.4) given in terms of two different auxiliary variables ~ and r. The same remark is valid for the solutions of Eqs. (3.8) and (3.19). Therefore, the following relations take place
F(m,~) = ~'(m, r),
~ ( m , ( ) = ~ ( m , r).
(3.20)
306
Shubov THEOREM
3.2 The following estimates hold < C[ral-x,
(3.21)
[L(,',,, r) + e-"+''"-~]e ~'-''''+~ < Clml-',
(3.22)
[~'(r,',,r)- r
< CI"I-',
(3.23)
[~',.(m,r)-sinhr]e-"
0.
(3.55)
Now let e~ = ~:l[n1-1, where ~:1 > 0 is a constant, which will be chosen later. Let B ~ . ( r ~ ) and OB~,,(~,,) be the disk and the circle of the radius e~ centered at the point Pn~. We estimate both sides of Eq. (3.54) on the circles OB~,,(Pn~,). Assume that m = Pn, + e~e i~ 9 OB,.(Cn,~), ~o 9 [0, 2~r]. Taking into account an explicit expression (3.53) for r ~ we obtain
ic(m)i < r
+ e.~'~l -~ = e(2A4)-' ln(c~+r :x) + A/" -
+ :::':I-' -< 2 ~ @ l n l ) - '
(Inl + 1/2).i
(1 + 0 (Inl-')) 9
(3.5~)
There exists no such that , - 1 ( 1 + O(Inl-')) < ~ for n >_ no. Therefore, from (3.56) we obtain
IC(m)l < r
for n > n0, m 9 0B,.(r~,).
(3.87)
Taking into account that {r~ =} are the roots of Eq. (3.53), we obtain the following estimate from below for the function on the left in Eq. (3.54) ~_e m'M-~ + c~+e-m'M+Ar = o~_e~',,-~l-Ar+e,,e~-~ + Cz+e-~'"J~+ff-e"e~'~[ a+e_ mo' ~ A 4 + A c ' + e ' ~ e 9* ~ M
--
a+e_,~.,~+ A / ' - e n e ' ~"A 4 ] ~ 2[a+e -'~-'~+ A t
[sinh(e,,e'~A/I)[ _> poe,,.M, where Po = 2 a+e - r
X
(3.58)
= 2 1 _ h21p(a ) 1/2. Now let us chose the constant ~:l in the
form ~:l = ~.~[-2Pol- We see that for n >_ nl (where nx is sufficiently large) the following estimates holds
~_~-z
+ ~+~-~+z
> It(re)l,
m 9
OB,.(~,),
(3.59)
312
Shubov
It follows from (3.59) and the Rouchet theorem that Eq. (3.51) has exactly one root in each disk B~ (Pn~) with n _< hi. The latter fact implies the asymptotic formula (2.10). To complete the proof of the theorem we have to derive the approximate formula for quasimodes. Let us return to the formula (3.49) for the generalized Jost solution and evaluate it when A = ,X,,. Denoting
we obtain the following formula for J(Am, z)
(1 +O(I,,I-')){,:,-e-""'" 0,(=))-88 (1 +
-
(3.61)
o(I,-,I-11) {h -z (lo(a)) -88coso$11(z).,.{-i (p(a)) 88sinO3ri(=) "lt- 0 (1"1-') } 9
Notice that
h -z (p(a)) -~ cosw,(x) + i (p(a))88sinw~(x) = A cos (w,~(x) - ~ ) ,
(3.62)
where A and ~ satisfy A c o s & = h -z(p(a))-{;
Asin~=i(p(a)) 88
(3.63)
Solving (3.63) for A and ~ we obtain = iln V/~-+a"-',
A 2 = (h -2 (p(a)) -1/2
-
(p(a)) 1/2)i/2.
Substituting these formulas into (3.62) we arrive at (2.9). Finally, we give an outline of the proof of the formulas (2.12), (2.13). To find the Jost function for the case h = 0 one should solve the system for the coefficients A(m) and B(m) which is similar to the system (3.36) and (3.37). The only difference consists of the following: instead of h -1 and m in the right-hand side of Eqs. (3.36) it should be 1 and 0 respectively. Carrying out all the steps as for the case h # 0 one arrives at the formulas (2.12), (2.13). Theorem 2.2 is proved. R E M A R K 3.3 To complete this section we note that: a) If Reh > 0, then all distant roots are located in a strip of the upper half plane ,X; if Reh < 0, then all distant roots are in a strip of the lower half-plane; both strips are parallel to the real axis. b) If Reh = 0, then we see, that IrnA,, - AP ---+0 when in[ ~ oo. It means, that ImA,, does not depend on p(a) asymptotically.
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4.
313
Double completeness of the set of eigenmodes and completeness of the sets ~+ and ~_. This section is devoted to the proof of Theorem 2.4. We divide this proof into
three steps. On the first step we introduce the energy space 7~ of two-component functions and defined an auxiliary matrix differential operator ~h on 9(. We show that ~h has a compact inverse (even ~7~1 E ~p for any p > 1), and the system of its root vectors is complete in 7-(. In this proof we use certain results of our work [4] and a well known Keldysh theorem (see below). The operator ~h is closely related to the pencil Lh(,X) introduced in Section 2 (see Remark 2.2, a)). The sets of eigenvalues of Th and of Lh(A) coincide and the root vectors of ~h can be expressed explicitly in terms of the eigenvectors and associated vectors of the pencil
Lh(,~). It turns out that the completeness of the root vectors of ~h is equivalent
to the so-called double completeness of the whole set ~ = 5+ t2 ~r_ of the eigenvectors and associated vectors of the pencil
Lh()~) (see Remarks 2.2, b) and c)). We do not use this fact
in our proof of Theorem 2.4. However, it is important in itself, and we discuss it briefly at the end of this section (see Remark 4.2, c)). Recall that on the first step of the proof we do not impose any restrictions on the multiplicities of the roots of the Jost function. On the second step of the proof we assume that h is real and that all roots of the Jost function are simple, i.e. the pencil Lh(~) has no associated vectors. Using the explicit
Lh(,~), we find the system biorthogonal to the
relation between the eigenvectors of Xh and system of eigenvectors of ~:h.
On the third step we combine the results of the first two steps and complete the proof.
Step 1. On this step we assume that h E C U {oo}. Let 7-~ be the Hilbert space obtained as a closure of two-component functions U = ( u0 ~ with respect to the following \ ul / energy norm
To be more precise, we consider the space 7-/as the space of equivalence classes, identifying U and V if and only if ul = vl and u o - v o =
corot. In 7"/we consider the differential operator 0
s
= -i
1
(4.2)
d2
p(z) with the domain =
{g En:
E
e
314
Shubov
ul(O) = O, u'o(a) + hu~(a) = 0~, if h
#
(4.3)
If h = ~ , the second boundary condition in (4.3) should be replaced by ul(a) = 0. The function p(z) in (4.2) is exactly the density function from our original problem (1.5)- (1.7). Let :D be the bounded operator defined by
= -i
(0 0)
,
(4.4)
0
where d(z) is the damping function from our original problem (1.5) - (1.7). Together with /::h we consider another operator ~:h defined on the same domain (4.3) by the formula ~h = / : h + :D. We note, that the operator s
(4.5)
has been studied in our papers [4, 5, 6]. We formulate below
(see Theorem 4.2) some properties of this operator which will be used in this section. Based on the properties of/:h we will prove that the set of root vectors of ~h is complete in 7-/. THEOREM
4.1 Let h e C 13 { ~ } and assume that p(z) and d(x) satisfy the
conditions (2.1) and (2.3). Then the system of root vectors of the operator ~h is complete in the energy space 7"[. REMARK
4.1 Notice that in Theorem 4.1 we do not use the condition (2.2).
The reason for this is the fact that in the proof of this theorem we do not use the asymptotic formulas of Theorem 2.2. The latter formulas make sense only if (2.2) is satisfied. (In fact, the asymptotic formulas for ~,, and F,~ can be derived in the case p(a - O) = Ihl. But in this case they are different from (2.10) and (2.11).) In other words, the completeness property of the set of root vectors of ~:h does not depend on the condition (2.2). However, if (2.2) is not satisfied then the R-basis property of the sets 5+ and ~_, which we prove in Section 5, will be destroyed. In the proof of completeness of 5+ and 5 - given in this section, we use the Rbasis property of the eigenvectors of/:h, and, therefore, we have to use (2.2). However, in fact, the completeness of~+ and 5 - can be shown by a different method without using (2.2). This will be done in our next paper, where we treat the general case of multiple roots of the Jost function. To prove Theorem 4.1 we have to recall the following corollary of the results of our work [4].
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315
T H E O R E M 4.2 [4]. For any h E CU{cr the following statements hold. 1) f-h has purely discrete spectrum. Its resolvent is a meromorphic function, all poles of which are simple, except maybe, for a finite number of them of a finite multiplicity each. 2) F.h~ exists and is a compact operator of the class 6~(7"~) for any p > 1. (We recall that ~p(7"l), 0 < p < ~ is the class of all compact operators A in 7"[, which have the following property: the sequence of the eigenvalues of the operator (AA*) 1/2 belongs to ~. By ~r we denote the class of all compact operators in 7"[ [2]).
3) The set of all root vectors of s forms an R-basis in 7"[. (We recall that a nonzero vector is called a root vector of the operator A corresponding to the eigenvalue )~o i f ( A - ~ o I ) ~ = 0 for some positive n (see [1, 2]). 4) For h ~ 0 the operator f.h 1 is a rank-one perturbation of s 1: ~'h 1 = ~-,01 -~
Ah,
(4.6)
where Ah is a one dimensional operator in 7-[. P r o o f o f T h e o r e m 4.1. First of all we can see that zero does not belong to the discrete spectrum of ~h. Indeed, using a contradiction argument, let us assume that there
existsan~176176
ET"lsuchthat~hG=O'gl
Itiseasytosee,
thatthe
latter equation implies (p(x))-tg~'(x) - 2d(X)gl(x) = O, g,(x) = 0, g~(a) + hgl(a) = 0. Due to gl(x) = 0, we have g'o'(X) = O, g~(a) = 0. We see that the only solution that satisfies the equation and the boundary conditions is go =
Const, gl -- 0. Under our agreement this
solution is considered as trivial. Based on Statement 2 of Theorem 4.2 we obtain
9 ;1 = L ; ' ( I + s),
(4.7)
where S E ~p, p > 1. Let us justify (4.7). Due to the existence of/:~-1 we have ~h = ( I + :Ds163
7)Z:h 1 E ~p, p > 1.
(4.8)
Due to the fact that ( - 1 ) is not an eigenvalue of the compact operator :D/:~-1 (because 0 is not an eigenvalue of ~:h) we have (I + Vs
-1 = I + S,
where
S = -V(I +
V)-1,
(4.9)
with V = :D/:~1 . Combining (4.8) and (4.9) we obtain (4.7). Now we use Statement 4 of Theorem 4.2. Substituting (4.6) into (4.7) we get
9 ;~ = (I + Ahs163 where T E ~p, p > 1. Due to the fact that Z:o = s the Keldysh theorem (see [1, 19]).
+ S),
(4.10)
we can apply the following corollary of
316
Shubov
T H E O R E M 4.3 [19] Let A be an operator in a Hilbert space H such that A =
(I + S I ) B ( I + S2), where B = B* 9 ~p, p < oo and S~,$2 9 6r
If the operator A vanishes
only on the zero vector, then the system of its root vectors is complete in H.
If we identify ~:~1, s Ahs and A, B, $1, $2, then from Theorem 4.3 we obtain the desired completeness of the root vectors of the operator ~ 1 . Theorem is proved. Step 2. On this step we assume that all roots of the Jost function are simple, and that h E RU{or It follows from the latter fact that the set of all roots is symmetric with respect to the imaginary axis: A_,~ = -X,, if Re An ~ O, n E Z'. Recall that the functions F,(x) = J(A=, z) are the eigenfunctions of the pencil Lh(A). It is easy to check that the operator ~:h has the same set of eigenvalues {A~,, n e Z'} as the pencil L,,(A) and the eigenvector of ~h corresponding to the eigenvalue A,~ has the form ,
r
~
neZ',
x e [ 0 , a].
(4.11)
f.(x)
To justify (4.11)consider the equation (~:h- A,I)~ = 0 with ~ ( z ) = ( f o ( x ) I \
fx(~) .
Rewriting
the latter equation in the component-wise form, we obtain (P(z)) -1 fo' - 2d(x)fx = i~,fx,
fa = i)~,,fo,
fl(0) = 0, f~(a) + hfl(a) = 0.
(4.12) (4.13)
It follows immediately from (4.12), (4.13) that f0 is the solution of the problem (1.5) - (1.7) with A = )~,, and therefore, f0 = (/A,~)-XF,. Thus, the formula (4.11) is shown. L E M M A 4.1 There ezists a unique system { ~ , n E Z'} biorthogonal to the system (4.11). This system is given by the formula
9 :(x)=
(
~
1
F-"(x)) Z '. F_,(x) , ne
(4.14)
Proof. Consider the operator ~, adjoint to ~:h- We know that ~ x E ~p(~) for p > 1. Therefore, ~ , has a purely discrete spectrum {~,, n E Z'} of simple eigenvalues. The set of eigenvectors of ~ , is exactly the system { ~ , n E Z'} biorthogonal to (4.11). So, we have to find these eigenvectors. It is a straightforward computation to check that ~=-i
(0 1) 1 ~ p(~) d~
2e(~,)
'
(4.15)
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317
o,~,--{o--('~) fl
Let ~
f~(a) - hf1(a)
E 7"/: f1(0) = 0,
}
= 0 .
(4.16)
be the eigenvector of ~:~ corresponding to the eigenvalue ~n. Reasoning exactly like
in the deriwtion of (4.11), we obtain
~:(~) =
~
F,:(x)
, n e Z',
where Fg is the solution of the problem (1.5) - (1.7) with I = - ~ , . ~
satisfies (1.5)- (1.7) with t = )~,~, and, therefore, F'~(x) =
F,(x).
(4.17) T h e latter means that Now recall that, since
h 9 Ri.J{~} the Jost function has the symmetry: J(A,x) = J ( - ~ , x ) , Using the latter two facts we obtain
F_,~(x).
F,~(x)
and, ~, = - A _ , .
= F-'~,(x) = J(A,, x) = J ( - ~ , , x) = J(A_,, x) =
Finally, we notice that the uniqueness of the biorthogonal system follows from the
completeness of the original system (4.11). The lemma is shown. S t e p 3. P r o o f o f T h e o r e m
2.4. We prove this Theorem using the contradic-
L~(O,a). T h e (f, F,,)L2(o,,) = 0 for n
tion argument. Assume, that the set ~+ is not complete in
latter fact means
the existence of a nontrivial f 9 L~(0, a) such that
= 1, 2 , . . . . Let us
define the function g by the formula
g(~) = oo~ c ~ e~i~, ~ ,
t~t,
dt
y(~)p(r)dt.
~ ~ 0 , o ~ o ~ , ~e~o~o~o, ~o voctor ( ~ ) ~
(4.1S) ~. ~ o ~ e
show, that
Indeed, we have
1
1
"
-
(L F.)L;(o,~] = ~ [ - i-~-:. ]~ f(~)p(~)dtr.(x) 0 +
2 fo~ Due to the boundary condition for
+ (S,F~)L~C0,o~] F,,(x)
(4.20)
at x = 0, we obtain
(4.21)
318
Shubov
Thus based on the assumption about f we obtain the orthogonality of the nontrivialvector ( ~ ) E ~ t o t h e s e t { ~ , , } ~ > l .
Thismeans, that due to Theorem 4.1, the
vector ( ~ ) must belong to the closure of the span of the biorthogonal vectors {~,~}~: forms an R-boris in L~(O, a). P r o o f . We have to show that every f 6 L20(0,a) can be uniquely represented in the form
f = ~_, h,,G,~.
(5.15)
n>_l
Let f be an arbitrary function from L~(0, a). Let F be an odd continuation of f on the interval [ - a , 0]. F can be uniquely decomposed with respect to the R-basis
{~.(~)}.~z, F(x) = Y~. c,,Q,(z) = - y]~ c ~ , , ( - x ) .
nEZ'
nEZ'
(5.16)
We see that for x > 0 the following formula holds for any even function g
Applying (5.17) to d(x) and/~(x) we have
where f14 and A/" are given by (2.4), (2.5). Using (5.14) and (5.18), we obtain for n > 0
Qn(-x)= 'p(x)]-'14exp(iA,~(2.A/[- f ' v l ~ d t )
~a d(,)~-~d,-~-In ~ } + v::+ o:' +
+ 2.hl" -
: (p(x))-l/4exp { - iAn ~ a ~/-~d' -
+ -:}
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323
Now based on (5.2) we evaluate i./~.4,, + N = i~r(n + 1/2) - In V/~+~ "t.
(5.20)
Substituting (5.20) into (5.19) we obtain
+ 2ni(n + 1/2) -
In
21n
a~+a:l~
= - (p(x))-1/4exp{--'.(x)}, (5.21)
where
=,.~ r
+[
+ 1o o:1
For n < 0 using (5.17) we obtain
~n(-x) = (p(x)) -U` exp{---Inl(-x)} =
--
(p(x)) -U'l e x p {---Inl(x)}"
(5.23)
Using (5.22) and (5.23) for the formula (5.16) we have for x > 0
(P(Z))I/4F(x) = E ~e':'"(~) + E C'ne-F'Inl(x)------{-- E (zne-='~(x) -- Z G'*eEl'*I(z)} (5.24) n>l
nl
na is minimal in L~(O,a). The latter is obvious, because this system can be obtained from an R-basis {~,~},ez' in two steps: selecting all functions with n _ 1 and restricting them to the interval [0,a]. Both steps preserve the minimal property of the system. The proof is complete. R E M A R K 5.1 If we take an even continuation of f , then we will have an unconditionally convergent series with respect to the corresponding "sinh" R-basis. S t e p 3. Now we will prove the R-basis property of the system {~(x)}~ez, in the space
L?;(-a, a). Let
324
Shubov
Then ~ ( z ) can be written in the form ~,(x) ----O,~(x)exp (r
sgn n ) .
(5.28)
In what follows we prove: a) the system of exponentials { & ( x ) ) , e Z' forms an R-basis in L~(-a, a); b) the multiplication of the above system by exp (r sgn n) cannot destroy the R-basis property. First we prove the statement b). A
L E M M A 5.2 /f (~,,(x)),~z, /s
an R-basis in L~-a,a), then {~,~(x)},ez, is,
also, an R-basis in L~-a, a). P r o o f . Assume, that {~"~}~ez, is an R-basis in g =
n~(-a, a). Let us introduce
the linear operator B defined on finite linear combinations of the vectors {~}~ez' by the formula a,O, B* E a,~a,,.
(5.29)
rt
It is sufficient to show that B can be extended to H as a bounded and boundedly invertible operator. Denote by 92l+ and 9Yt- the closed linear spans of {~}~>0 and {~,}~ 0 we have:
~ = (~(x)) -t/" exp{(iTrM-l(n + 1/2) - M -t In ~ + a - '
- M-'X) f"
~-~dt},
(5.30)
= (ff(~))-U' exp((-i~rYk4-1(rn q- 1/2) q- .h.4-1 In V/'~+a-' -t- .A~-I-Af)fxa ~C~dt}-(5-31) Using
(5.30)
and (5.31)andintroducing a new independent variable y = [ "
~ / ~ d t 9 [0, 2A4],
A
we obtain after a straightforward computation that ( ~ , ~-,~)hr = 0 for n, m > 0. Denote by A the operator of multiplication by the function exp{r Clearly, A is bounded and boundedly invertible. Notice, that (5.28) can be rewritten in the form ~ = A~,,
0-,~ = A-I~-,~,
n, m > 0.
(5.32)
We see, that the subspaces A f l ~ and A-~ffYt- are spanned by {g,}~>0 and {g,)~
