From Lieb-Thirring inequalities to spectral enclosures for the damped wave equation
aa r X i v : . [ m a t h . SP ] A ug From Lieb–Thirring inequalities to spectral enclosures forthe damped wave equation
David Krejˇciˇr´ık and Tereza Kurimaiov´a
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering,Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republicdavid.krejcirik@fjfi.cvut.cz & kurimter@fjfi.cvut.cz
12 August 2020
Abstract
Using a correspondence between the spectrum of the damped wave equation and non-self-adjoint Schr¨odinger operators, we derive various bounds on complex eigenvalues of the former.In particular, we establish a sharp result that the one-dimensional damped wave operator issimilar to the undamped one provided that the L norm of the (possibly complex-valued)damping is less than 2. It follows that these small dampings are spectrally undetectable. Consider a classical or a quantum system described by the damped wave equation u tt + a ( x ) u t − ∆ x u = 0 (1)in the space-time variables ( x, t ) ∈ R d × (0 , ∞ ), where the ‘damping’ a is a complex-valued function.The positive (respectively, negative) part of the real part of a corresponds to the dissipation(respectively, excitation) of mechanical or electromagnetic waves, while the imaginary part of a can be interpreted as a conservative perturbation of a Dirac (quasi-)particle.Writing U := ( u, u t ) T , it is customary to replace the scalar second-order equation (1) by thefirst-order evolution system U t = A a U with the matrix-valued damped wave operator A a := (cid:18) − a (cid:19) , dom A a := H ( R d ) × ˙ H ( R d ) , (2)acting in the Hilbert space H := ˙ H ( R d ) × L ( R d ). A careful analysis of the stationary problem A a Ψ = µ Ψ in H , where µ ∈ C is a spectral parameter, provides information on the behaviour ofthe time-dependent solutions u of (1). It is easy to see that the spectral problem is related to theoperator pencil (see Lemma 1 below) S µa ψ := − ∆ ψ + µ a ψ = − µ ψ in L ( R d ) . (3)As an example of usefulness of the spectral data, let us recall the collaboration of the firstauthor with P. Freitas [19], where it is shown that the damped wave equation (1) becomes unstablewhenever the real-valued damping a achieves a sufficiently negative minimum. The strategy of [19]is based on establishing spectral asymptotics of the family of Schr¨odinger operators S µa as µ → + ∞ . Consequently, A a possesses a real positive point µ in the spectrum, which is responsible fora global instability of (1).The present paper is partially motivated by a remark of T. Weidl [30] that Lieb–Thirringinequalities known for S µa could provide more quantitative information on the location of the real spectrum of A a . Moreover, in view of the unprecedented interest in non-self-adjoint Schr¨odinger1perators in the near past, new complex extensions of the Lieb–Thirring inequalities have beenderived in recent years. Consequently, the T. Weidl’s observation has become interesting for the complex spectrum of A a as well. In this paper we go beyond the usual setting by considering even complex-valued dampings.The literature on damped wave systems is rather extensive and we restrict ourselves on quotingthe following recent works and refer the interested reader for further references therein. Therelationship between the damped wave and Dirac equations is discussed on an abstract levelin [22] and related one-dimensional Lieb-Thirring-type inequalities can be found in [8, 9]. Anextensive spectral analysis of the wave operator with possibly unbounded damping is performedin [20]. Basic eigenvalue bounds for weakly damped wave systems can be deduced from [18]. Theperturbation of eigenvalues of abstract damped wave operators has been recently studied in theframework of Krein spaces in [27]. Resolvent estimates for an abstract dissipative operator arederived in [4, 29].The structure of this paper is as follows. In Section 2 we comment on basic properties ofthe damped wave operator A a and state a crucial correspondence between its eigenvalue problemand the operator pencil (3). Self-adjoint and non-self-adjoint Lieb–Thirring-type inequalities areapplied in Sections 3 and 4, respectively. In particular, in Theorem 4 we prove that the pointspectrum of the one-dimensional damped wave operator A a is empty provided that k a k L ( R ) < A a is actually similar to the skew-adjoint undamped operator A under the same smallness condition. Although the damped wave equation can be made meaningful for certain unbounded dampings( cf. [20]), our standing hypothesis is that the damping a : R d → C is bounded, i.e. , a ∈ L ∞ ( R d ) . (4)Then it is easy to see that there exists a constant c such that A a + c is a generator of a C -semigroupof contractions ( cf. [19, App. B]), so (1) is well posed. Equivalently, A a is a quasi-m-accretiveoperator. In particular, the operator A a is closed, so its spectrum is well defined.Notice that A a is skew-adjoint ( i.e. , iA a is self-adjoint) if, and only if, a is purely imaginary.In particular, the undamped operator A is skew-adjoint. To have this symmetry result, it isimportant that we consider the homogeneous Sobolev space ˙ H ( R d ) (defined as the completion of C ∞ ( R d ) with respect to the norm ψ
7→ k∇ ψ k ) in the definition of the Hilbert space H , see (2).On the other hand, it is important to keep in mind that ˙ H ( R d ) is not a subset of L ( R d ).With an abuse of notation, we denote by − ∆ the distributional Laplacian as well as its self-adjoint realisation in L ( R d ) with domain dom( − ∆) := H ( R d ). For any bounded potential V : R d → C , the Schr¨odinger operator S V = − ∆ + V with dom S V = H ( R d ) is a well definedm-sectorial operator.We say that V is vanishing at infinity and write V ∞ −→
0, provided thatlim R → + ∞ k V k L ∞ ( R d \ B R (0)) = 0 , where B R (0) denotes the ball of radius R > V is a relatively compactperturbation of − ∆. Consequently, σ e ( S V ) = σ e ( S ) = [0 , + ∞ ) for any choice of the essentialspectrum (namely, σ e1 , . . . , σ e5 in the notation of [11, Chapt. IX]). More specifically, it is clearfor the components σ e1 , . . . , σ e4 , which are preserved by relatively compact perturbations, and theresult extends to the widest choice σ e5 because S V is m-sectorial ( cf. [24, Prop. 5.4.4]).It is easy to see that the spectrum of the undamped operator A is purely continuous and equalto the imaginary axis; in particular, σ ( A ) = σ e ( A ) = i R . If a is vanishing at infinity, then thedamping represents a relatively compact perturbation of A . Consequently, σ e k ( A a ) = σ e k ( A ) =2 R for k = 1 , . . . ,
4. To see that it is true also for σ e5 , it is enough to notice that each connectedcomponent of C \ i R intersects the resolvent set of A a due to (4). In summary, a ∞ −→ ⇒ σ e ( A a ) = σ e ( A ) = i R . Since the essential spectrum is independent of a , the present paper focuses on the point spec-trum of A a . The following lemma specifies the equivalence between the spectral problem for (2)and the operator pencil (3) in the case of eigenvalues (not necessarily discrete). Lemma 1.
Assume (4) . For every µ ∈ C , − µ ∈ σ p ( S µa ) ⇐⇒ µ ∈ σ p ( A a ) . Proof.
Assuming − µ ∈ σ p ( S µa ), there exists a non-trivial function ψ ∈ H ( R d ) such that S µa ψ = − µ ψ . Then Ψ := ( ψ , µψ ) T ∈ dom A a and( A a Ψ) T = ( µψ , ∆ ψ − aµψ ) = ( µψ , − S µa ψ ) = µ ( ψ , µψ ) = µ Ψ T . Conversely, assuming µ ∈ σ p ( A a ), there exists a non-trivial Ψ = ( ψ , ψ ) T ∈ dom A a such that A a Ψ = µ Ψ. In other words, ψ ∈ H ( R d ), ψ ∈ ˙ H ( R d ) and ψ = µψ , ∆ ψ − aψ = µψ .Combining the latter two equations, we get S µa ψ = − µ ψ with ψ = 0. In this section, we assume that the damping a satisfying (4) is real-valued and focus on real eigenvalues µ ∈ R . Recalling the correspondence of Lemma 1, it is enough to consider the auxiliarySchr¨odinger operators S V with real-valued potentials V . Then S V is self-adjoint, so its spectrumis purely real. Let us denote by { λ n ( V ) } Nn =1 the non-decreasing sequence of negative discreteeigenvalues of S V , where each eigenvalue is repeated according to its multiplicity. The set can beeither empty ( N = 0) or finite (1 ≤ N < ∞ ) or infinite ( N = ∞ ).Our starting point are the self-adjoint Lieb–Thirring inequalities ( cf. [26, 25]) N X n =1 | λ n ( V ) | γ ≤ L γ,d Z R d V γ + d − , (5)where L γ,d is a positive constant independent of V and V ± := ( | V | ± V ). More specifically, it isknown that such a bound holds with a finite constant L γ,d if, and only if, γ ≥ if d = 1 ,γ > d = 2 ,γ ≥ d ≥ . (6)The sharp values of the constants L γ,d are not known explicitly for all the admissible values of γ .In special cases, however, one knows that L , = and L γ,d = L cl γ,d := Γ( γ + 1)2 d π d Γ( γ + d + 1) for d ≥ , γ ≥ Theorem 1.
Suppose that a is real-valued and assume (4) . Let γ be any number satisfying (6) .If µ is a positive (respectively, negative) eigenvalue of A a and a − ∈ L γ + d ( R d ) (respectively, a + ∈ L γ + d ( R d ) ), then µ γ − d ≤ L γ,d Z R d a γ + d − (cid:18) respectively, ( − µ ) γ − d ≤ L γ,d Z R d a γ + d + (cid:19) . n the other hand, if a − ∈ L d ( R d ) (respectively, a + ∈ L d ( R d ) ) and Z R d a d − < L d ,d respectively, Z R d a d + < L d ,d ! , then A a has no positive (respectively, negative) eigenvalues.Proof. From Lemma 1 we get that real µ ∈ σ p ( A a ) if, and only if, there exists n ∈ N such that λ n ( µa ) = − µ . Hence, (5) implies µ γ = | λ n ( µa ) | γ ≤ L γ,d µ γ + d Z R d a γ + d − for µ ∈ σ p ( A a ) ∩ (0 , + ∞ ) , | µ | γ = | λ n ( µa ) | γ ≤ L γ,d | µ | γ + d Z R d a γ + d + for µ ∈ σ p ( A a ) ∩ ( −∞ , . (7)This proves the first part of the theorem. The case 0 ∈ σ p ( A a ) cannot occur because the spectrumof S is purely continuous. Dividing (7) by | µ | γ + d , and eventually putting γ = d , which can bedone for all d ≥
1, we conclude with the second part of the theorem.To continue, we restrict ourselves to d = 1. In this case the Buslaev–Faddeev–Zakharov traceformulae ( cf. [12]) provides us with a lower bound for the sum of square roots of the eigenvaluesof S µa , namely N X n =1 | λ n ( µa ) | ≥ − µ Z R a . (8)Of course, the inequality is non-trivial only if µ R R a <
0. The latter is known to be a sufficientcondition which guarantees that inf σ ( S µa ) <
0. Assuming in addition a ∞ −→
0, it follows that S µa possesses at least one negative eigenvalue. The number of eigenvalues N of S µa is controlled fromabove by the Bargmann bound ( cf. [28, Problem 22]) N ≤ | µ | Z R | a ( x ) || x | x. . Consequently, for µ satisfying the inequality | µ | < (cid:18)Z R | a ( x ) || x | x. (cid:19) − (9)the operator S µa has exactly one negative eigenvalue λ ( µa ) and we get from (8) the estimate | µ | = | λ ( µa ) | ≥ − µ Z R a . (10)This implies R R a ≥ − µ > R R a ≤ µ <
0. In summary, we have established thefollowing result.
Theorem 2.
Let d = 1 . Suppose that a is real-valued and in addition to (4) assume a ∈ L ( R , | x | x. ) and a ∞ −→ . Let µ be a real eigenvalue of A a . If µ > and R R a < − (or µ < and R R a > ),then | µ | ≥ (cid:18)Z R | a ( x ) || x | x. (cid:19) − . Finally, combining the Lieb–Thirring inequalities with the Buslaev–Faddeev–Zakharov traceformulae, we obtain the following quantitative bounds on the location of real eigenvalues of theone-dimensional damped wave operator. The presence of the coupling parameter α is useful forstudying the stability of solutions of (1) in the spirit of [17, 21, 19].4 heorem 3. Let d = 1 . Suppose that a is real-valued and in addition to (4) assume a ∈ L ( R , | x | x. ) and a ∞ −→ . Let R R a < (respectively, R R a > ). For any µ > (respectively, µ < ) satisfy-ing (9) , there exists exactly one α > satisfying (cid:18)Z R a − (cid:19) − ≤ α ≤ − (cid:18)Z R a (cid:19) − respectively, (cid:18)Z R a + (cid:19) − ≤ α ≤ (cid:18)Z R a (cid:19) − ! such that µα is an eigenvalue of A αa .Proof. Assuming µ > R R a < a ∞ −→ S µa possessesexactly one negative eigenvalue. Applying (7) with γ = and (10), we get − µ Z R a ≤ | λ ( µa ) | ≤ µ Z R a − . Similarly, for µ < R R a >
0, we get − µ Z R a ≤ | λ ( µa ) | ≤ | µ | Z R a + . These estimates together with Lemma 1 prove the theorem.
In this section, we use the recent progress in spectral theory of non-self-adjoint Schr¨odinger op-erators and state results about complex eigenvalues of the damped wave operator. There is noobstacle to consider complex-valued dampings at the same time.Let us start with dimension d = 1. The celebrated result of E. B. Davies et al. (see [1, Thm. 4]and [10, Corol. 2.16]) states that every eigenvalue λ ( V ) of S V with V ∈ L ( R ) satisfies the bound | λ ( V ) | ≤ Z R | V ( x ) | x. . (11)Moreover, the bound is known to be optimal for step-like potentials approximating the Diracpotential. Now, assuming in addition to (4) that a ∈ L ( R ), if µ is any eigenvalue of A a (necessarilyit must be non-zero, cf. the proof of Theorem 1), Lemma 1 ensures that there exists λ ( µa ) ∈ σ p ( S µa ) such that | µ | = | λ j ( µa ) | ≤ | µ | Z R | a ( x ) | x. , where the inequality is due to (11). Dividing by | µ | , it follows that the point spectrum of A a is empty provided that the L -norm of a is small, namely k a k L ( R ) <
2. Let us summarise theobservation into the following theorem.
Theorem 4.
Let d = 1 . In addition to (4) assume a ∈ L ( R ) . If k a k L ( R ) < , then σ p ( A a ) = ∅ .Moreover, the constant is optimal.Proof. It remains to argue about the optimality. Our strategy is to show that for any numberslightly greater than 2 there exists a damping W with the L -norm equal to this number suchthat A W has an eigenvalue. For this we choose the analytically computable case, where thedamping is a step-like potential W ( x ) := x < − b ,a if − b < x < b , x > b , with a < < b . A W Ψ = µ Ψ reduces to ∆ ψ − µW ψ − µ ψ = 0, where ψ is the firstcomponent of Ψ. It is enough to analyse the situation of real µ . For µ = 0 or µ = − a weget just a trivial solution to the eigenvalue equation. Also, using the spectral correspondence ofLemma 1, we know that all the real eigenvalues of A W must be positive, provided that W ≤ µ ≤ k a k L ∞ ( R ) = − a . Thus the only interval where we can find a real eigenvalue is (0 , − a ).Now, for ψ to lie in H ( R ) ⊂ C ( R ) and be non-trivial, it can be easily verified that the secularequation F ( µ ) := 2 p − ( µa + µ ) cos (cid:0) b p − ( µa + µ ) (cid:1) + ( a + 2 µ ) sin (cid:0) b p − ( µa + µ ) (cid:1) = 0must be satisfied. For µ ∈ (0 , − a ) this is equivalent to G ( µ ) := p − ( µa + µ ) F ( µ ) = 0. Wecompute lim µ → + G ( µ ) = 0 = lim µ →− a − G ( µ ) , lim µ → + G ′ ( µ ) = 2 a ( − c ) , lim µ →− a − G ′ ( µ ) = 2 a (1 + c ) , where c := − ba = k W k L ( R ) . We observe that for c > G ( µ ) are the same,implies that there exists µ ∗ ∈ (0 , − a ) such that G ( µ ∗ ) = 0. Hence, µ ∗ ∈ σ p ( A W ) which proves thedesired optimality. Remark 1.
Taking b := (2 | a | ) − and α ∈ C , the complexified step-like potential αW convergesin the sense of distributions to αδ as a → −∞ , where δ is the Dirac delta function. Replacing(formally) a by αδ in (3), one arrives at the pencil problem − ψ ′′ = − µ ψ in R \ { } ,ψ (0 + ) − ψ (0 − ) = 0 ,ψ ′ (0 + ) − ψ ′ (0 − ) = µα ψ (0) . (12)There exists no admissible solution ψ ∈ H ( R \ { } ), unless α = − α = 2) inwhich case every µ ∈ C with ℜ µ > ℜ µ <
0) is an ‘eigenvalue’ ! Since k αW k L ( R ) = | α | , this is another support for the optimality of the constant 2 in Theorem 4.However, we are not aware of any result about a spectral approximation of the operator pencil (3)by bounded potentials.The damped wave equation on a finite interval with the damping being a Dirac delta functionwas previously studied in [3], [2, Sec. 4.1.1] and [7].In Section 5, we argue that the absence of eigenvalues follows as a consequence of the similarityof A a to the undamped wave operator A , provided that k a k L ( R ) < d ≥
2, we use the robust result of R. Frank et al. (see [15, Thm. 1] and[16, Thm. 3.2]) stating that every eigenvalue λ ( V ) of S V with V ∈ L γ + d ( R d ), 0 < γ ≤ , satisfiesthe bound | λ ( V ) | γ ≤ D γ,d Z R d | V ( x ) | γ + d x. , (13)where D γ,d is a constant independent of V . Using Lemma 1, it follows that any eigenvalue µ of A a satisfies | µ | γ = | λ j ( µa ) | γ ≤ D γ,d | µ | γ + d Z R d | a ( x ) | γ + d x. . (14)We therefore conclude with the following theorem.6 heorem 5. Let d ≥ . In addition to (4) assume a ∈ L γ + d ( R d ) with < γ ≤ . There exists aconstant D γ,d such that, for any eigenvalue µ ∈ σ p ( A a ) , one has | µ | γ − d ≤ D γ,d Z R d | a | γ + d . The formal analogue of (13) for γ = 0 and d ≥ D ,d such that if D ,d Z R d | V ( x ) | d x. < , (15)then S V has no eigenvalues. The case of discrete eigenvalues is due to R. Frank [15, Thm. 2], whilepossibly embedded eigenvalues were covered by [16, Thm. 3.2]. Independently, the total absenceof eigenvalues under a weaker hypothesis for d = 3 and alternative conditions in higher dimensionswas obtained by L. Fanelli, D. Krejˇciˇr´ık and L. Vega in [13] (see also [14] and [6, 5]). In fact, thespectral stability of S V for potentials V which are small in some sense goes back to the abstractresult of T. Kato’s [23] that we shall recall for other purposes in the following section. Here wejust mention that a straightforward combination of Lemma 1 and (15) implies that (14) holds alsofor γ = 0 and d ≥
3. Consequently, we get the following formal analogue of Theorem 5.
Theorem 6.
Let d ≥ . In addition to (4) assume a ∈ L d ( R d ) . There exists a positive con-stant D ,d such that, for any eigenvalue µ ∈ σ p ( A a ) , one has | µ | − d ≤ D ,d Z R d | a | d . In this section, we come back to the one-dimensional setting of Theorem 4. We find the resultappealing because it implies that small dampings cannot be distinguished from the undampedsystem just by measuring eigenfrequencies. In fact, the following result shows a much strongerresult that small dampings are indeed spectrally undetectable . Theorem 7.
Let d = 1 . In addition to (4) assume a ∈ L ( R ) . If k a k L ( R ) < , then theoperators A a and A are similar to each other. Moreover, the constant is optimal. Here the similarity means that there exists an operator W ∈ B ( H ) such that W − ∈ B ( H )and A a = W A W − . In other words, iA a is quasi-self-adjoint ( cf. [24]), because iA is self-adjoint.Then the spectra of A a and A coincide. In particular, A a must have the same eigenvalues as A .Since the spectrum of A is purely continuous, Theorem 4 follows as a direct consequence ofTheorem 7. Proof.
The optimality of the constant 2 follows from the proof of Theorem 4. Indeed, for anynumber strictly larger than 2, there exists a damping a whose L -norm equals that number,while A a possesses eigenvalues. This would violate the similarity.To prove the similarity, we use the abstract result of T. Kato [23, Theorem 1.5]. Writing iA a = iA + ( − i sgn a B ) ∗ B with B := (cid:18) | a | (cid:19) , where sgn is the complex sign (defined by sgn f := f / | f | if f = 0 and sgn f := 0 if f = 0), it isenough to show that the bounded operators B and − i sgn a B are relatively smooth with respectto iA ( cf. [23, Def. 1.2]) andsup ξ ∈ C \ R k K ξ k < , where K ξ := BR ( ξ, iA ) Bi sgn a
7s the Birman–Schwinger operator. Here we use the notation R ( ξ, T ) := ( T − ξ ) − for the resolventof an operator T at point ξ ∈ C \ σ ( T ). The relative smoothness is a rather complicated conditionin general, but it reduces to reasonable criteria when iA is self-adjoint ( cf. [23, Thm. 5.1]).Using in addition the simple structure of the intertwining operators B and − i sgn a B in our case,everything reduces to verifying the unique conditionsup ξ ∈ C \ R k ˜ K ξ k < , where ˜ K ξ := BR ( ξ, iA ) B . (16)It can be easily verified that R ( ξ, i A ) = R ( ξ , − ∆) (cid:18) ξ ii ∆ ξ (cid:19) . Consequently, ˜ K ξ = | a | ξR ( ξ , − ∆) | a | (cid:18) (cid:19) , and therefore k ˜ K ξ k = | ξ | (cid:13)(cid:13)(cid:13) | a | R ( ξ , − ∆) | a | (cid:13)(cid:13)(cid:13) , where k · k denotes the operator norm both in H and L ( R ) on the left- and right-hand side,respectively. The latter will be estimated by the Hilbert–Schmidt norm k · k HS . To this purpose,we recall the explicit formula for the integral kernel of R ( z, − ∆): G z ( x, y ) = e −√− z | x − y | √− z , where x, y ∈ R , z ∈ C \ [0 , + ∞ ) and the principal branch of the square root is used. Using theelementary estimate | G z ( x, y ) | ≤ p | z | , we therefore get k ˜ K ξ k ≤ | ξ | (cid:13)(cid:13)(cid:13) | a | R ( ξ , − ∆) | a | (cid:13)(cid:13)(cid:13) = | ξ | Z R × R | a ( x ) | | G ξ ( x, y ) | | a ( y ) | x. y. ≤ k a k L ( R ) , where the last bound is independent of ξ . Recalling (16), T. Kato’s similarity condition [23,Theorem 1.5] holds provided that k a k L ( R ) < Acknowledgment
The research was partially supported by the GACR grants No. 18-08835S and 20-17749X.
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