Spectral Properties of Schrödinger-type Operators and Large-time Behavior of the Solutions to the Corresponding Wave Equation

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1 Math. Model. Nat. Phenom. Vol. 8, No., 23, DOI:.5/mmn/2386 Sectral Proerties of Schrödinger-tye Oerators and Large-time Behavior of the Solutions to the Corresonding Wave Equation A.G. Ramm Deartment of Mathematics, Kansas State University, Manhattan, KS , USA Abstract. Let L be a linear, closed, densely defined in a Hilbert sace oerator, not necessarily selfadjoint. Consider the corresonding wave equations () ẅ +Lw =, w() =, ẇ() = f, ẇ = dw dt, f H. (2) ü+lu = fe ikt, u() =, u() =, where k > is a constant. Necessary and sufficient conditions are given for the oerator L not to have eigenvalues in the half-lane Rez < and not to have a ositive eigenvalue at a given oint k 2 d >. These conditions are given in terms of the large-time behavior of the solutions to roblem () for generic f. Sufficient conditions are given for the validity of a version of the iting amlitude rincile for the oerator L. A relation between the iting amlitude rincile and the iting absortion rincile is established. Keywords and hrases: ellitic oerators, wave equation, iting amlitude rincile, iting absortion rincile Mathematics Subject Classification: 35P25, 35L9, 43A32. Introduction Let L be a linear, densely defined, closed oerator in a Hilbert sace H. Our results and techniques are valid in a Banach sace also, but we wish to think about L as of a Schrödinger-tye oerator in a Hilbert sace and, at times, think that L is selfadjoint. For a Schrödinger oerator L = 2 +q(x) the resolvent (L k 2 ),Imk >, is an integral oerator with a kernel G(x,y,k), its resolvent kernel. If q is a real-valued function, sufficiently raidly decaying then L is selfadjoint, G(x, y, k) is analytic with resect to k in the half-lane Imk >, excet, ossibly, for a finitely many simle oles ik j, k j >, the Corresonding author. ramm@math.ksu.edu c EDP Sciences, 23 Article ublished by EDP Sciences and available at htt:// or htt://dx.doi.org/.5/mmn/2386

2 semiaxis k is filled with the oints of absolutely continuous sectrum of L, and there exists a it G(x,y,k +iǫ) = G(x,y,k) ǫ for all k >. Sufficient conditions for k 2 = not to be an eigenvalue of L are found in aers [5], [6]. Sectral analysis of the Schrödinger oerators is resented in many books (see, for examle, [2] and []). In aers [3], [4], such an analysis was given in a class of domains with infinite boundaries aarently for the first time, see also [8]. In [7] an eigenfunctions exansion theorem was roved for non-selfadjoint Schrödinger oerators with exonentially decaying comlex-valued otential q. The oerator L in this aer is not necessarily assumed to be selfadjoint. In [] the validity of the iting amlitude rincile for some class of selfadjoint oerators L has been established. This rincile says that, as t, the solution to roblem has the following asymtotics ü+lu = fe ikt, u() =, u() =, u = du dt, (.) where k is a real number and v H solves the equation u = e ikt v +o(), t, (.2) Lv k 2 v = f. (.3) Thev iscalled the iting amlitude. Itturnsoutthat amorenaturaldefinition oftheiting amlitude is: t v = u(s)e iks ds, (.4) t t if this it exists and solves equation (.3). Why is this definition more natural than (.2)? There are good reasons for this. One of the reasons is: if (.2) and (.3) hold, then the it (.4) exists and solves equation (.3). The other reason is: the it (.4) may exist and solve equation (.3) although the it (.2) does not exist. Examle. If u = e ikt v + e ikt v, then the it (.2) does not exist, while the it (.4) does exist and is equal to v. To describe our assumtions and results, some rearation is needed. Consider the roblem ẅ+lw =, w() =, ẇ() = f. (.5) Assuming that u(t) ce at, where c > stands throughout the aer for various generic constants, and a is a constant, one can define the Lalace transform of u(t), U := U() := e t u(t)dt, σ > a, where = σ +iτ, Re = σ. Let us take the Lalace transform of (.) and of (.5) to get LU + 2 U = f +ik, (.6) and LW + 2 W = f, (.7) 28

3 where W = W() = w(t)e t dt. We also denote W() := w(t). The comlex lane is related to the comlex lane k by the formula = ik, k = k +ik 2, k 2, σ = k 2. (.8) We assume throughout that f is generic in the following sense: If I is the identity oerator and a oint is a ole of the kernel of the oerator (L+ 2 I), then it is a ole of the same order of the element (L+ 2 I) f = W. Ifk 2 isaneigenvalueoflandrek 2 iarg k2 <, thenimk >, wherek = k e 2, = ik, soσ = Re >. Let k > and assume that k 2 < is an eigenvalue of L. Then ik is a ole of the resolvent kernel G(x,y,k), and = i(ik) = k is a ole of the kernel of the oerator (L + 2 I). If k 2 > is an eigenvalue of L, then = ik is a ole of the oerator (L+ 2 I). The following known facts from the theory of Lalace transform will be used. Proosition.. An analytic in the half-lane σ > σ function F() is the Lalace transform of a function f(t), such that f(t) = for t < and f(t) 2 e 2σt dt < (.9) if and only if su F(σ +iτ) 2 dτ <. (.) σ>σ Proosition.2. If F() = f(t), then F() = f(s)ds. (.) Let us now formulate the main Assumtions A and B standing throughout this aer. Assumtion A. For a generic f the W() = (L+ 2 ) f is analytic in the half-lane σ >, excet, ossibly, at a finitely many simle oles at the oints ik j, j J, k j are real numbers, and at the oints κ m, Reκ m >, W() = J j= v j +ik j +W ()+ M m= b m κ m, (.2) where v j and b m are some elements of H, W () is an analytic function in the half-lane Re = σ >, continuous u to the imaginary axis σ =, and satisfying the following estimate Assumtion B. There exists the it for all real numbers k. W () c + γ, γ > 2. (.3) W (σ ik) W ( ik) = (.4) σ 29

4 Theorem.3. Let the Assumtion A hold. Then a necessary and sufficient condition for the oerator L to have no eigenvalues in the half-lane Rek 2 < is the validity of the estimate w(s)ds = O(eǫt ), t, (.5) for an arbitrary small ǫ >. A necessary and sufficient condition for the oerator L not to have any ositive eigenvalues k 2 > is the validity of the estimate t e iks w(s)ds t = o(), t, k R. (.6) A oint ik >,k >, is not a ole of the resolvent kernel of the oerator (L k 2 i) if and only if estimate (.6) holds with k = k >. Remark. If condition (.6) holds for k =, then w(s)ds = o(t), so condition (.5) holds, and the oerator L has no eigenvalues in the half-lane Rek 2 <. Theorem.4. Let the Assumtions A and B hold. Suose that estimates (.4) and (.5) hold. Then the iting amlitude rincile (.4) holds for every k R, k k j, j J. In section 2, roofs are given. 2. Proofs 2.. Proof of Theorem.3 From the Assumtion A and Proosition., it follows that W() is a Lalace transform of a function w(t) such that J M w(t) = v j e ikjt + b m e κmt +w (t), (2.) where j= w (t) = m= σ+i σ i e t W ()d, (2.2) and the integral in (2.2) converges in L 2 -sense due to the assumtion (.3). It is clear from formula (2.) that all b m = if and only if estimate (.5) holds with < ǫ < min m M Reκ m. This roves the first conclusion of Theorem.3. Let us calculate the exression on the left side of formula (.6) and show that this exression is o() unless k = k j for some j J. In this calculation it is assumed that L does not have any eigenvalues in the half-lane Rek 2 <, in other words, that all b m =. Otherwise the exression on the left of formula (.6) tends to infinity as t at an exonential rate. If all b m = in (2.), then J j= v j t If k and k j are real numbers, then e i(k kj)t dt+ t w (t)e ikt dt := I +I 2. (2.3) t t e i(k kj)t dt = 2 {, k = kj,, k k j. (2.4)

5 Thus, I = if and only if k does not coincide with any of k j, j J. Let us rove that t w (t)e ikt dt =. (2.5) t t By roosition (.2) and the Mellin inversion formula, one has I := t w (t)e ikt dt = σ+i σ i W ( ik) et d, (2.6) t where Re = σ > can be chosen arbitrarily small. Let t = q, take σ = t, write q = +is, and write the integral on the right side of (2.6) as: I = +i i W ( q t ik)q t e q q2dq. (2.7) If one uses estimate (.3) and formula q = (+s 2 ) /2, then one obtains the following inequality I 2πt (+s 2 ) /2 cds [+ +is t ik γ ] = c 2πt γ Let s = ty. Then the integral on the right side of (2.8) can be written as ct 2πt γ = c 2π c 2π dy (+t 2 y 2 ) /2 (t γ +[+t 2 (y k) 2 ] γ/2 ) dy (+t 2 y 2 ) /2 (+[t 2 +(y k) 2 ] γ/2 ) ds (+s 2 ) /2 (t γ +[+(s kt) 2 ] γ/2 ). (2.8) dy (+t 2 y 2 ) /2 [+(y k) γ, as t, (2.9) ] and the convergence of the last integral to zero is uniform with resect to k R. Thus I =. (2.) t From (2.3)-(2.5) the last two conclusions of Theorem.3 follow. Theorem.3 is roved Proof of Theorem.4 Using Proosition.2, equation (.6), and the Mellin formula, one gets where, according to (.6), Let σ = t and t = q. Then t t u(t)e ikt dt = t σ+i σ i U( ik) e t d, (2.) U( ik) = W( ik). (2.2) u(t)e ikt dt = +i i ( q ) e q W t ik q2dq. (2.3) Estimate (.5) and Theorem.3 imly that all b m = in formula (2.). Therefore, using formula (2.) with b m =, one gets J W = v j +W, +ik j j= 2

6 and ( q ) ( q ) W t ik = W t ik + J j= v j q t i(k k j). (2.4) One has t n = n! n+. Therefore and Furthermore, t v +i j i t +i e q i q2dq =, e q q t i(k k j) q 2dq = +i i W ( q t ik ) e q { ivj k k j, k k j,, k = k j. (2.5) q 2dq = W ( ik), (2.6) as follows from assumtion (.4) and the Lebesgue s dominated convergence theorem if one asses to the it t under the sign of the integral (2.6). Let us check that this v solves equation (.3). This would conclude the roof of Theorem.4. We need a lemma. Lemma 2.. If h L loc (, ) and the it t t h(s)ds exists, then the it e t h(t)dt exists, and Proof of Lemma. One has For any > one has t t h(s)ds = e t h(t)dt. (2.7) e t h(t)dt = e t h(s)ds + 2 te t t h(s)dsdt. e t h(s)ds =. Let q = t and denote H(t) := t h(s)ds, J := t H(t). Then 2 te t t h(s)dsdt = qe q H(q )dq. Passing in the last integral to the it one obtains (2.7). Lemma is roved. Using equation (2.7), one writes v = U( ik), where U solves equation (.6). Thus, LU( ik)+( ik) 2 U( ik) = f. Multilying both sides of this equation by and assing to the it, one obtains equation (.3). In the assage to the it under the sign of the unbounded oerator L the assumtion that L is closed was used. Thus, the conclusion of Theorem.4 follows. If the it (.4) exists at a oint = iτ then one says that the iting absortion rincile holds for the oerator L at the oint k = i = i( ik) = k,k >. Thus, Assumtion B means that the iting absortion rincile holds for L at the oint k >, that is, ǫ (L k 2 iǫ) f exists. 22

7 3. Alications Let L = 2 +q(x), where q(x) is a real-valued function, q(x) c(+ x ) 2 ǫ,ǫ >,x R 3. Then L is selfadjoint on the domain H 2 (R 3 ). Its resolvent (L k 2 i) satisfies Assumtions A and B if one kees in mind the following. Let G(x,y,k) be the resolvent kernel of L, that is, the kernel of the oerator (L k 2 i), LG(x,y,k) = δ(x y) in R 3, G L 2 (R 3 ) for Imk >. If f L 2 (R 3 ) is comactly suorted, then for k > the function v(x) := (L k 2 i) f = G(x,y,k)f(y)dy R 3 does not necessarily belong to L 2 (R 3 ). For examle, if q(x) =, then G(x,y,k) = eik x y 4π x y, and the function ( ) v(x,k) = g(x,y,k)dy = O x y (3.) does not belong to L 2 (R 3 ) (excet for those k > for which x(x,k) = in the region y. These numbers k > are the zeros of the Fourier transform of the characteristic function of the ball y, see [], Chater. By this reason the abstract results of theorem (.3) and (.4) can be used in alications if one defines some subsace of H, for examle, a subsace of functions with comact suort, denote by P, a rojection oerator on this subsace, and relaces W and W by PW and PW in equations (.2) and (.4). For examle, the function (3.) one relaces by η(x)v(x, k), where η(x) is a characteristic function of a comact subset of R 3. Theanalyticroertiesofη(x)v(x,k)andofv(x,k)asfunctionsofk arethesame. Asimilarsuggestion is used in []. With the above in mind, one knows (for examle, from [2] or []) that Assumtions A and B hold for L = 2 +q(x). Consequently, the conclusions of Theorems.3 and.4 hold. In addition, the assumtions q(x) c(+ x ) 2 ǫ, ǫ >, Imq =, imly that L does not have ositive eigenvalues, so all v j =, and zero is not an eigenvalue of L if ǫ > (see [5], [6]). A new method for estimating of large time behavior of solutions to abstract evolution roblems is develoed in [9], where some alications of this method are given. References [] D. Eidus. The iting amlitude rincile. Russ. Math. Surveys, 24, (969), 5 94 [2] D. Pearson. Quantum scattering and sectral theory. Acad. Press, London, 988. [3] A.G. Ramm. Sectral roerties of the Schrödinger oerator in some domains with infinite boundaries. Doklady Acad of Sci. USSR, 52, (963) [4] A.G. Ramm. Sectral roerties of the Schrödinger oerator in some infinite domains. Mat. Sborn., 66, (965), [5] A.G.Ramm. Sufficient conditions for zero not to be an eigenvalue of the Schrödinger oerator. J. Math. Phys., 28, (987), [6] A.G.Ramm. Conditions for zero not to be an eigenvalue of the Schrödinger oerator. J. Math. Phys. 29, (988), [7] A.G. Ramm. Eigenfunction exansion for nonselfadjoint Schrödinger oerator. Doklady Acad. Sci. USSR. 9, (97),

8 [8] A.G. Ramm. Scattering by obstacles. D.Reidel, Dordrecht, 986. [9] A.G. Ramm.Stability of solutions to some evolution roblems. Chaotic Modeling and Simulation (CMSIM),, (2), [] A.G. Ramm. Inverse roblems. Sringer, New York, 25. [] M. Schechter. Oerator methods in quantum mechanics. North Holland, New York,