Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation
aa r X i v : . [ m a t h . SP ] M a y EIGENVALUE ASYMPTOTICS, INVERSE PROBLEMSAND A TRACE FORMULA FOR THE LINEARDAMPED WAVE EQUATION
DENIS BORISOV AND PEDRO FREITAS
Abstract.
We determine the general form of the asymptotics forDirichlet eigenvalues of the one–dimensional linear damped waveoperator. As a consequence, we obtain that given a spectrum corre-sponding to a constant damping term this determines the dampingterm in a unique fashion. We also derive a trace formula for thisproblem. Introduction
Consider the one–dimensional linear damped wave equation on theinterval (0 , w tt + 2 a ( x ) w t = w xx + b ( x ) w, x ∈ (0 , , t > w (0 , t ) = w (1 , t ) = 0 , t > w ( x,
0) = w ( x ) , w t ( x,
0) = w ( x ) , x ∈ (0 , u xx − ( λ + 2 λa − b ) u = 0 , x ∈ (0 , , (1.2) u (0) = u (1) = 0 , (1.3)and has received quite a lot of attention in the literature since the pa-pers of Chen et al. [CFNS] and Cox and Zuazua [CZ]. In the first ofthese papers the authors derived formally an expression for the asymp-totic behaviour of the eigenvalues of (1.2), (1.3) in the case of a zeropotential b , which was later proved rigorously in the second of the above Date : November 17, 2018.2000
Mathematics Subject Classification.
Primary 35P15; Secondary 35J05.D.B. was partially supported by RFBR (07-01-00037) and gratefully acknowl-edges the support from Deligne 2004 Balzan prize in mathematics. D.B. isalso supported by the grant of the President of Russia for young scientist andtheir supervisors (MK-964.2008.1) and by the grant of the President of Russiafor leading scientific schools (NSh-2215.2008.1) P.F. was partially supported byFCT/POCTI/FEDER. . papers. Following this, there were several papers on the subject which,among other things, extended the results to non–vanishing b [BR], andshowed that it is possible to design damping terms which make thespectral abscissa as large as desired [CC]. In [F2] the second author ofthe present paper addressed the inverse problem in arbitrary dimen-sion giving necessary conditions for a sequence to be the spectrum ofan operator of this type in the weakly damped case. As far as weare aware, these are the only results for the inverse problem associatedwith (1.2), (1.3). Other results for the n − dimensional problem include,for instance, the fact that in that case the decay rate is no longer de-termined solely by the spectrum [L], a study of some particular caseswhere the role of geometric optics is considered [AL], the asymptoticbehaviour of the spectrum [S] and the study of sign–changing dampingterms [F1].The purpose of the present paper is twofold. On the one hand,we show that problem (1.2), (1.3) may be addressed in the same wayas the classical Sturm–Liouville problem in the sense that, althoughthis is not a self–adjoinf problem, the methods used for the formerproblem may be applied here with similar results. This idea was alreadypresent in both [CFNS] and [CZ]. Here we take further advantage ofthis fact to obtain the full asymptotic expansion for the eigenvaluesof (1.2), (1.3) (Theorem 1). Based on these similarities, we were alsoled to a (regularized) trace formula in the spirit of that for the Sturm–Liouville problem (Theorem 4).On the other hand, the idea behind obtaining further terms in theasyptotics was to use this information to address the associated inversespectral problem of finding all damping terms that give a certain spec-trum. Our main result along these lines is to show that in the case ofconstant damping there is no other smooth damping term yielding thesame spectrum (Corollary 2). Namely, we obtain the criterion for thedamping term to be constant. Note that this is in contrast with theinverse (Dirichlet) Sturm–Liouville problem, where for each admissiblespectrum there will exist a continuum of potentials giving the samespectrum [PT]. In particular this result shows that we should expectthe inverse problem to be much more rigid in the case of the waveequation than it is for the Sturm–Liouville problem. This should beunderstood in the sense that, at least in the case of constant damping,it will not be possible to perturb the damping term without disturbingthe spectrum, as is the case for the potential in the Sturm–Liouvilleproblem.The plan of the paper is as follows. In the next section we set thenotation and state the main results of the paper. The proof of the IGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 3 asymptotics of the eigenvalues is done in Sections 3 and 4, where inthe first of these we derive the form of the fundamental solutions ofequation (1.2), while in the second we apply a shooting method tothese solutions to obtain the formula for the eigenvalues as zeros of anentire function – the idea is the same as that used in [CZ]. Finally, inSection 5 we prove the trace formula.2.
Notation and results
It is easy to check that if λ is an eigenvalue of the problem (1.2),(1.3), then λ is also an eigenvalue of the same problem. In view of thisproperty, we denote the eigenvalues of this problem by λ n , n = 0, andorder them as follows . . . Im λ − Im λ − Im λ Im λ . . . while assuming that λ − n = λ n . We also suppose that possible zeroeigenvalues are λ ± = λ ± = . . . = λ ± p = 0. If p = 0, the problem(1.2), (1.3) has no zero eigenvalues. For any function f = f ( x ) wedenote h f i := R f ( x ) d x . Theorem 1.
Suppose a ∈ C m +1 [0 , , b ∈ C m [0 , , m > . Theeigenvalues of (1.2), (1.3) have the following asymptotic behaviour as n → ±∞ : λ n = πn i + m − X j =0 c j n − j + O ( n − m ) , (2.1) were the c j ’s are numbers which can be determined explicitly. In par-ticular, c = −h a i , c = h a + b i π i , (2.2) c = 12 π (cid:20) h a ( a + b ) i − h a ih a + b i + a ′ (1) − a ′ (0)2 (cid:21) . (2.3)A straightforward consequence of the fact that the spectrum deter-mines the average as well as the L norm of the damping term (as-suming b fixed) is that the spectrum corresponding to the constantdamping determines this damping uniquely. Corollary 2.
Assume that a ∈ C [0 , , λ n are the eigenvalues of theproblem (1.2), (1.3), the function b ∈ C [0 , is fixed, and the formula(2.1) gives the asymptotics for these eigenvalues. Then the function a ( x ) is constant, if and only if c = 2 π i c − h b i , DENIS BORISOV AND PEDRO FREITAS in which case a ( x ) ≡ − c . In the same way, the asymptotic expansion allows us to derive otherspectral invariants in terms of the damping term a . However, thesedo not have such a simple interpretation as in the case of the aboveconstant damping result. Corollary 3.
Suppose b ≡ , a i ( x ) = a ( x ) + e a i ( x ) , i = 1 , , where a (1 − x ) = a ( x ) , e a i (1 − x ) = − e a i ( x ) , e a i , a ∈ C [0 , , and for a = a i the problems (1.2), (1.3) have the same spectra. Then h e a i = h e a i , h a e a i = h a e a i is valid. From Theorem 1 we have that the quantity Re( λ n − c ) behaves as O ( n − ) as n → ∞ . This means that the series ∞ X n = −∞ n =0 ( λ n − c ) = 2 ∞ X n =1 Re( λ n − c )converges. In the following theorem we express the sum of this series interms of the function a . This is in fact the formula for the regularizedtrace. Theorem 4.
Let a ∈ C [0 , , b ∈ C [0 , . Then the identity ∞ X n = −∞ n =0 ( λ n − c ) = a (0) + a (1)2 − h a i holds. Asymptotics for the fundamental system
In this section we obtain the asymptotic expansion for the fundamen-tal system of the solutions of the equation (1.2) as λ → ∞ , λ ∈ C . Thisis done by means of the standard technique described in, for instance,[E, Ch. IV, Sec. 4.2, 4.3], [Fe, Ch. II, Sec. 3].We begin with the formal construction assuming the asymptotics tobe of the form(3.1) u ± ( x, λ ) = e ± λx ± x R φ ± ( t,λ ) d t , where(3.2) φ ± ( x, λ ) = m X i =0 φ ( ± ) i ( x ) λ − i + O ( λ − m − ) , m > . IGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 5
In what follows we assume that a ∈ C m +1 [0 , b ∈ C m [0 , λ . It leads us to a recurrent system ofequations determining φ ( ± ) i which read as follows: φ ( ± )0 = a, (3.3) φ ( ± )1 = −
12 ( ± a ′ + a + b ) , (3.4) φ ( ± ) i = − ± φ ( ± ) i − ′ + i − X j =0 φ ( ± ) j φ ( ± ) i − j − ! , i > . (3.5)The main aim of this section is to prove that there exist solutions to(1.2) having the asymptotics (3.1), (3.2). In other words, we are goingto justify these asymptotics rigorously. We will do this for u + , the caseof u − following along similar lines.Let us write U m ( x, λ ) = e λx + m P i =0 λ − i x R φ (+) i ( t ) d t . In view of the assumed smoothness for a and b we conclude that U m ∈ C [0 , U ′′ m − λ U m − λaU m + bU m = λ − m e λx f m ( x, λ ) , x ∈ [0 , ,U m (0) = 1 , U ′ m (0) = λ + m X i =0 φ (+) i (0) λ − i , where the function f m satisfies the estimate | f m ( x, λ ) | C m uniformly for large λ and x ∈ [0 , λ >
0. Differentiating the function u + formally we see that u ′ + (0 , λ ) = λ + φ + (0 , λ ) = λ + m X i =0 φ (+) i (0) λ − i + O ( λ − m − ) . Let A ( λ ) = λ + m X i =0 φ (+) i (0) λ − i , and u + ( x, λ ) be the solution to the Cauchy problem for the equation (cid:7) (1.2) subject to the initial conditions u + (0 , λ ) = 1 , u ′ + (0 , λ ) = A ( λ ) . DENIS BORISOV AND PEDRO FREITAS
We introduce one more function w m ( x, λ ) = u + ( x, λ ) /U m ( x, λ ). Thisfunction solves the Cauchy problem( U m w ′ m ) ′ + λ − m U m e λx f m w m = 0 , x ∈ [0 , ,w m (0 , λ ) = 1 , w ′ m (0 , λ ) = 0 . The last problem is equivalent to the integral equation w m ( x, λ ) + λ − m ( K m ( λ ) w m )( x, λ ) = 1 , ( K m ( λ ) w m )( x, λ ) := x Z U − m ( t ) t Z U m ( t )e λt f m ( t , λ ) w m ( t , λ ) d t d t . Since Re λ > > t > t >
1, the estimate | U − m ( t , λ ) U m ( t , λ )e λt | C m holds true, where the constant C m is independent of λ , t , t . Hence,the integral operator K m : C [0 , → C [0 ,
1] is bounded uniformly in λ large enough, Re λ >
0. Employing this fact, we conclude that w m ( x ) = 1 + O ( λ − m ) , λ → ∞ , Re λ > , in the C [0 , φ + ( x, λ ) = m − X i =0 φ (+) i ( x ) λ − i + O ( λ − m ) , gives the asymptotic expansion for the solution of the Cauchy problem(1.2), (3.6) as λ → ∞ , Re λ > λ
0. Let A ( λ ), A ( λ ) be functions havingthe asymptotic expansions A ( λ ) = λ + m X i =0 λ − i Z φ (+) i ( x ) d x, A ( λ ) = λ + m X i =0 φ (+) i (1) λ − i . We define the function e u + ( x, λ ) as the solution to the Cauchy problemfor equation (1.2) subject to the initial conditions e u + (1 , λ ) = e A ( λ ) , e u ′ + (1 , λ ) = A ( λ )e A ( λ ) . In a way analogous to the arguments given above, it is possible tocheck that the function e u + has the asymptotic expansion (3.1) in the C [0 , λ → + ∞ , Re λ
0. Hence, e u + (0 , λ ) = 1 + O ( λ − m )for each m >
1. In view of this identity we conclude that the function u + ( x, λ ) := e u + ( x, λ ) / e u + (0 , λ ) is a solution to (1.2), satisfies the condi-tion u + (0 , λ ) = 1, and has the asymptotic expansion (3.1), where theasymptotics for φ + is given in (3.7). IGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 7
For convenience we summarize the obtained results in
Lemma 3.1.
Let a ∈ C m +1 [0 , , b ∈ C m [0 , . There exist two linearindependent solutions to the equation (1.2) satisfying the initial con-dition u ± (0 , λ ) = 1 and having the asymptotic expansions (3.1) in the C [0 , -norm as λ → ∞ , λ ∈ C , where φ ± ( x, λ ) = m − X i =0 φ ( ± ) i ( x ) λ − i + O ( λ − m ) . Asymptotics of the eigenvalues
This section is devoted to the proof of Theorem 1 and Corollaries 2and 3. We assume that a ∈ C m +1 [0 , b ∈ C m [0 , m > u = u ( x, λ ) be the solution to (1.2) subject to the initial condi-tions u (0 , λ ) = 0, u ′ (0 , λ ) = 1. Denote γ ( λ ) := u (1 , λ ). The function γ is entire, and its zeros coincide with the eigenvalues of the problem(1.2), (1.3). It follows from Lemma 3.1 that, for λ large enough thefunction u ( x, λ ) can be expressed in terms of u ± by u ( x, λ ) = u + ( x, λ ) − u − ( x, λ ) u ′ + (0 , λ ) − u ′− (0 , λ ) . The denominator is non-zero, since due to (3.1) u ′ + (0 , λ ) − u ′− (0 , λ ) = 2 λ + 2 h a i + O ( λ − ) , λ → ∞ . Thus, for λ large enough(4.1) γ ( λ ) = u + (1 , λ ) − u − (1 , λ ) u ′ + (0 , λ ) − u ′− (0 , λ ) . Lemma 4.1.
For n large enough, the set Q := { λ : | Re λ | < πn + π/ , | Im λ | < πn + π/ } contains exactly n eigenvalues of the problem (3.1), (3.3).Proof. Let γ ( λ ) := γ ( λ )e λ + h a i . The zeros of γ are those of γ ( λ ). For λ large enough we represent the function γ ( λ ) as (cid:7) γ ( λ ) = γ ( λ ) + γ ( λ ) , γ := e λ + a (0)) − λ + a (0)) ,γ ( λ ) = − γ ( λ ) e φ + (0 , λ ) + e φ − (0 , λ ) + 2(1 + λ − a (0))(1 − e λ − h e φ + ( · ,λ ) i )2 λ ( λ + a (0)) + e φ + (0 , λ ) + e φ − (0 , λ )+ e λ − h e φ + ( · ,λ ) i − e − λ − h e φ − ( · ,λ ) i λ + a (0)) + λ − ( e φ + (0 , λ ) + e φ − (0 , λ )) , DENIS BORISOV AND PEDRO FREITAS e φ ± ( x, λ ) := λ − ( φ ± ( x, λ ) − a ( x )) . It is clear that for λ large enough the function γ ( λ ) satisfies an uniformin λ estimate | γ ( λ ) | C | λ | − (cid:0) | γ ( λ ) | + 1 (cid:1) . One can also check easily that (cid:7) | γ ( λ ) | > C | λ | , λ ∈ ∂K, if n is large enough. These two last estimates imply that | γ ( λ ) | | γ ( λ ) | as λ ∈ ∂K , if n is large enough. By Rouch´e theorem we concludethat for such n the function γ has the same amount of zeros inside Q as the function γ does. Since the zeros of the latter are given by πn i − h a i , n = 0, this completes the proof. (cid:3) Proof of Theorem 1.
Assume first that a ∈ C [0 , b ∈ C [0 , γ ( λ ) = 0. It follows from Lemma 4.1 that theseeigenvalues tend to infinity as n → ∞ . By Lemma 3.1, for λ largeenough the equation γ ( λ ) = 0 becomese λ + h φ + ( · ,λ )+ φ − ( · ,λ ) i = 0which may be rewritten as(4.2) 2 λ + h φ + ( · , λ ) + φ − ( · , λ ) i = 2 πn i , n ∈ Z . If we now replace φ ± by the leading terms of their asymptotic expan-sions we obtain 2 λ + 2 h a i + O ( λ − ) = 2 πn i , (4.3) λ = πn i − h a i + o (1) , n → ∞ . Hence, the eigenvalues behave as λ ∼ πn i − h a i for large n . Moreover,it follows from Lemma 4.1 that it is exactly the eigenvalue λ n whichbehaves as λ n = πn i − h a i + o (1) , n → ∞ . It follows from this identity and (4.3) that λ n = πn i − h a i + O ( n − ) , n → ∞ , and we complete the proof in the case m = 1. If m = 2, we substitutethe above identity and (3.1) into (4.2) and get λ n + h a i + 1 λ n h φ (+)1 + φ ( − )1 i + O ( λ − n ) = πn i ,λ n = πn i − h a i − h φ (+)1 + φ ( − )1 i πn i + O ( n − ) . IGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 9
The last formula and the identities (3.4) yield formulas (2.2) for c and c . Repeating the described procedure one can easily check that theasymptotics (2.1), (2.2) hold true. (cid:3) Proof of Corollary 2.
The coefficients c , c in the asymptotics (2.1)are determined by the formulas (2.2) and, by the Cauchy-Schwarz in-equality, we thus obtain c = h a i h a i = 2 π i c − h b i , with equality if and only if a ( x ) is a constant function. This factcompletes the proof. (cid:3) Proof of Corollary 3.
It follows from (2.2), (2.3) that h a i = h a i , h a i = h a i . Now we check that h a i i = h a i + h e a i i , h a i i = h a i + 3 h a e a i i , i = 1 , , and arrive at the statement of the theorem. (cid:3) Regularized trace formulas
In this section we prove Theorem 4. We follow the idea employed inthe proof of the similar trace formula for the Sturm-Liouville operatorsin [LS, Ch. I, Sec. 14].We begin by defining the functionΦ( λ ) := λ p ∞ Y n = p +1 (cid:18) − λλ n (cid:19) (cid:18) − λλ n (cid:19) . The above product converges, since (cid:18) − λλ n (cid:19) (cid:18) − λλ n (cid:19) = 1 + λ − λ Re λ n | λ n | , and by Theorem 1 we have(5.1) | λ n | = π n − π i c + c + O ( n − ) , Re λ n = c + O ( n − )as n → + ∞ . Proceeding in the same way as in the formulas (14.8),(14.9) in [LS, Ch. I, Sec. 14], we obtainΦ( λ ) = C Ψ( λ ) sinh λλ , Ψ( λ ) := ∞ Y n =1 (cid:18) − π n − | λ n | + 2 λ Re λ n π n + λ (cid:19) , C := ( πn ) p ∞ Y n = p +1 π n | λ n | . In what follows we assume that λ is real, positive and large. In thesame way as in [LS, Ch. I, Sec. 14] it is possible to derive the formula(5.2) ln Ψ( λ ) = − ∞ X k =1 k ∞ X n =1 (cid:18) π n − | λ n | + 2 λ Re λ n π n + λ (cid:19) k . Our aim is to study the asymptotic behaviour of ln Ψ( λ ) as λ → + ∞ .Employing the same arguments as in the proof of Lemma 14.1 and inthe equation (14.11) in [LS, Ch. I, Sec. 14], we arrive at the estimate ∞ X n =1 (cid:18) π n − | λ n | + 2 λ Re λ n π n + λ (cid:19) k c k λ k ∞ X n =1 π n + λ ) k c k λ k + ∞ Z d t ( π t + λ ) k = c k λ k + ∞ Z d z ( π z + 1) k c k +1 λ k , ∞ X k =3 k ∞ X n =1 (cid:18) π n − | λ n | + 2 λ Re λ n π n + λ (cid:19) k = O ( λ − ) , λ → + ∞ , (5.3)where c is a constant independent of k and n . Let us analyze theasymptotic behaviour of the first two terms in the series (5.2). As k = 1, we have(5.4) ∞ X n =1 π n − | λ n | + 2 λ Re λ n π n + λ = ∞ X n =1 π n − | λ n | − π i c + c π n + λ + (cid:0) π i c − c + 2 λc (cid:1) ∞ X n =1 π n + λ + 2 λ − S − λ − ∞ X n =1 π n (Re λ n − c ) π n + λ , where S := ∞ X n =1 (Re λ n − c ) . Taking into account (5.1), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 π n − | λ n | − π i c + c π n + λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ∞ X n =1 n ( π n + λ )= π λ − λλ Cλ − , IGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 11 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 (Re λ n − c ) π n π n + λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ∞ X n =1 π n + λ Cλ − , where the constant C is independent of λ . Here we have also used theformula(5.5) ∞ X n =1 π n + λ = λ coth λ − λ = λ − − λ − O ( λ − e − λ )as λ → + ∞ . We employ this formula to calculate the remaining termsin (5.4) and arrive at the identity(5.6) ∞ X n =1 π n − | λ n | + 2 λ Re λ n π n + λ = c + (cid:18) S − c − c πc (cid:19) λ − + O ( λ − ) , as λ → + ∞ . For k = 2 we proceed in the similar way, ∞ X n =1 (cid:18) π n − | λ n | + 2 λ Re λ n π n + λ (cid:19) = ∞ X n =1 ( π n − | λ n | ) ( π n + λ ) − λ ∞ X n =1 ( π n − | λ n | ) Re λ n ( π n + λ ) + 4 λ c ∞ X n =1 π n + λ ) + 4 λ ∞ X n =1 (Re λ n ) − c ( π n + λ ) . By differentiating (5.5) we obtain ∞ X n =1 π n + λ ) = λ coth λ − − λ (1 − coth λ )4 λ . This identity and (5.1) yield that as λ → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 ( π n − | λ n | ) ( π n + λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ∞ X n =1 π n + λ ) Cλ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 ( π n − | λ n | ) Re λ n ( π n + λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ∞ X n =1 π n + λ ) Cλ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 (Re λ n ) − c ( π n + λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ∞ X n =1 π n + λ ) Cλ − . Hence,(5.7) ∞ X n =1 (cid:18) π n − | λ n | + 2 λ Re λ n π n + λ (cid:19) = c λ − + O ( λ − ) as λ → + ∞ . It follows from (5.2), (5.3), (5.6), (5.7) thatln Ψ( λ ) = − c − (2 S − c + i πc ) λ − + O ( λ − ) , Φ( λ ) = C e − c sinh λλ (cid:2) − (2 S − c + i πc ) λ − + O ( λ − ) (cid:3) (5.8)as λ → + ∞ . It follows from (4.1) and Lemma 3.1 that for λ largeenough the estimate (cid:7) | γ ( λ ) | C | λ | − e | λ | holds true. Hence, the order of the entire function γ ( λ ) is one. In viewof Theorem 1 we also conclude that the series ∞ P n = p +1 | λ n | − convergesand therefore the genus of the canonical product associated with γ isone. We apply Hadamard’s theorem (see, for instance, [Le, Ch. I, Sec.10, Th. 13]) and obtain that γ ( λ ) = e P ( λ ) Φ( λ ) , P ( λ ) = α λ + α + 2 ∞ X n = p +1 | λ n | − Re λ n , where α , α are some numbers. Hence, due to (5.8), it follows that γ behaves as γ ( λ ) = C e P ( λ ) sinh λλ (cid:2) − (2 S − c + i πc ) λ − + O ( λ − ) (cid:3) , as λ → + ∞ . On the other hand, Lemma 3.1 and (4.1) imply that γ ( λ ) = e λ + h a i λ (cid:2) h φ (+)1 i − a (0)) λ − + O ( λ − ) (cid:3) + O ( λ − e − λ ) , as λ → + ∞ . Comparing the last two identities yields α = 0, C e α − c +2 ∞ P n =1 | λ n | − Re λ n = e h a i and − (2 S − c + i πc ) = h φ (+)1 i − a (0) . It now follows from (2.2), (3.4) that ∞ X n = −∞ n =0 ( λ n − c ) = 2 S = c + a (0) − h φ (+)1 i − i πc = a (0) + a (1)2 − h a i , completing the proof of Theorem 4. Acknowledgments
This work was done during the visit of D.B. to the Universidade deLisboa; he is grateful for the hospitality extended to him. P.F. wouldlike to thank A. Laptev for several conversations of this topic.
IGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 13
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