Electronic Journal of Differential Equations, Vol. 20052005), No. 45, pp. 9. ISSN: 072-669. URL: http://ejde.math.tstate.edu or http://ejde.math.unt.edu ftp ejde.math.tstate.edu login: ftp) ZEROS OF THE JOST FUNCTION FOR A CLASS OF EXPONENTIALLY DECAYING POTENTIALS DAPHNE GILBERT, ALAIN KEROUANTON Abstract. We investigate the properties of a series representing the Jost solution for the differential equation y + q)y = λy, 0, q LR + ). Sufficient conditions are determined on the real or comple-valued potential q for the series to converge and bounds are obtained for the sets of eigenvalues, resonances and spectral singularities associated with a corresponding class of Sturm-Liouville operators. In this paper, we restrict our investigations to the class of potentials q satisfying q) ce a, 0, for some a > 0 and c > 0. We consider the differential equation. Introduction y + q)y = λy for 0,.) where q LR + ) is real or comple-valued, with the boundary condition y0) cosα) + y 0) sinα) = 0 for some α [0, π)..2) In this paper, we consider the consequences of changes on the potential q rather than on the boundary condition.2) and we therefore restrict ourself to the classical case α [0, π). For an analysis of Sturm-Liouville operators with real valued, eponentially decaying potentials and nonselfadjoint boundary conditions, see for eample [6]. Let z = λ, Imz) > 0. Since q LR + ), there eists a unique L 2 R + )-solution χ, z) of.) satisfying χ, z) = e iz + o)) as +, which is known as the Jost solution [3]. Let φ, z 2 ) be the solution of.) satisfying φ0, z 2 ) = 0, φ 0, z 2 ) =. Then φ, z 2 ) satisfies.2) with α = 0 and we have W 0 χ, z), φ, z 2 ) ) = χ0, z), Imz) > 0, where W 0 denotes the Wronskian evaluated at = 0. Note that φ, z 2 ) and χ, z) are linearly dependent if and only if χ0, z) = 0 for some z such that Imz) > 0. The non-zero eigenvalues of the operator L 0 associated with.) and the Dirichlet 2000 Mathematics Subject Classification. 34L40, 35B34, 35P5, 33C0. Key words and phrases. Jost solution; Sturm-Liouville operators; resonances; eigenvalues; spectral singularities. c 2005 Teas State University - San Marcos. Submitted October 4, 2005. Published December 8, 2005.
2 D. GILBERT, A. KEROUANTON EJDE-2005/45 boundary condition are therefore of the form λ = z 2, where z is a zero of the Jost function χz) = χ0, z) satisfying Imz) > 0. If q is real-valued these zeros are situated on the segment line z = it, 0 < t < +, giving rise to negative eigenvalues. Moreover, if q is eponentially decaying, i.e. if q satisfies q) = Oe a ) as +.3) for some a > 0 then, whether q is real or comple-valued, the Jost function χz) can be analytically etended to the half plane {z C : Imz) > a/2} [9, 0, appendi II] and the part of the epansion in generalised eigenfunctions related to the continuous spectrum contains a spectral-type function of the form π z χz)χ z) ), z > 0..4) The epansion in eigenfunctions and generalised eigenfunctions in the case of eponentially decaying, comple-valued potentials was established by Naimark [9]. If q is real-valued the spectral-type function.4) is actually the spectral density associated with L 0 since, in this case, χ z) = χz) for Imz) = 0. The latter was proved by Kodaira [8] for a real-valued potential q. If we set χ π/2, z) = d d χ, z) and χ π/2z) = χ π/2 0, z), then the non-zero eigenvalues of the operator L α associated with.) and.2) are of the form λ = z 2, where z is a zero of χ α z) satisfying Imz) > 0, with χ α, z) = χ, z) cosα) + χ π/2, z) sinα) and χ α z) = χ α 0, z)..5) To see this note that χ, z) and φ α, z 2 ) are linearly dependent if and only if χ α z) = 0, where φ α, z 2 ) is a solution of.) satisfying.2), more precisely φ α 0, z 2 ) = sinα), φ 0, z 2 ) = cosα). If q satisfies.3), then χ α z) can be analytically etended to the half-plane {Imz) > a/2} [9, 0, appendi II]. It is then likely that the zeros of χ α z) situated just below the real ais will affect the behaviour of.4) [2, 4, 5]. Such a zero is called a resonance and, if q is real valued and if the zero is situated on the semi-ais it, 0 < t < +, it is said to be an antibound state. For Imz) = 0, z 0, we also have [0, appendi II] W 0 χ α, z), χ α, z)) = 2iz, so that χ α z) and χ α z) cannot vanish at the same time for Imz) = 0, z 0. If q is real-valued, then χ α z) = χ α z) and the equality above implies that χ α z) cannot vanish for Imz) = 0, z 0. On the other hand, if q is comple-valued, then χ α z) can vanish for some z with Imz) = 0. If z is such a zero of χ α z), then λ = z 2 is called a spectral singularity. The form of the epansion in generalised eigenfunctions obtained by Naimark [9, 0, appendi II] depends on whether such spectral singularities do eist. If there is no spectral singularity, then the epansion takes a form similar to that obtained by Kodaira [8]. It is to be noted that, for q LR + ), there are no L 2 R + )-solutions of.) for λ > 0 so that the spectral singularities cannot be associated with L 2 R + )-solutions
EJDE-2005/45 ZEROS OF THE JOST FUNCTION 3 of.). Moreover, if q also satisfies.3), then the number of spectral singularities is finite [9, 0, appendi II]. The literature available on the study of eigenvalues, resonances and spectral singularities is already abundant but we propose here an alternative method that allows us to view them as a single mathematical object, namely as arising from the zeros of the Jost function. Our method is relatively simple and allows us, in particular, to investigate resonance-free regions for eponentially decaying potentials. More detailed results are obtained on the set of resonances for compactly supported and super-eponentially decaying potentials in [4, 5] and in [2] for a class of eponentially decaying potentials. The relationship between the Jost function and the classical Titchmarsh-Weyl function is briefly outlined in section 5. 2. The Series It was shown by Eastham [, 2] that, for a real-valued integrable potential q, the Jost solution χ, z) can be represented in the form 2.). However, it is not difficult to show that the results below also hold when q is comple-valued and integrable. We have χ, z) = e iz + ) r n, z), 2.) with r 0, z) =, Also, with r n, z) = i 2z s n, z) = 2 From 2.2) we have so that, for Imz) > 0, qt)r n t, z) e 2izt )) dt, n. 2.2) d d χ, z) = eiz iz + ) s n, z), 2.3) r, z) = i 2z qt)r n t, z) + e 2izt ) )dt n. 2.4) r 0, z) =, qt) e 2izt )) dt r, z) z It is readily seen by induction on n that 0 qt) dt. r n, z) q ) n, n 0, 0, Imz) > 0, z where is the LR + )-norm, from which it follows that + r n, z) q ) n. z n 0
4 D. GILBERT, A. KEROUANTON EJDE-2005/45 The series in 2.) therefore converges absolutely and uniformly for 0, Imz) > 0 and z > q. Note that we supposed only that q LR + ). This result is similar to the one obtained by Rybkin [, theorem 3.]. We now investigate the convergence of 2.) for a class of eponentially decaying potentials. 3. Main Results We suppose throughout this section that q) ce a, 0, 3.) holds for some c > 0 and a > 0. We first consider the case α = 0 and then eamine the case α 0, π). In the latter case the details get rather cumbersome but, since we are aware of only few results concerning this case, we mention it anyway. Let δ > 0 and let Λ a,δ = {z C : Imz) > a/3, z > δ}. Lemma 3.. Suppose that 3.) holds and fi δ > /a. Then r n, z) ) ne na, 0, Imz) > a/3, n n! z a and the series 2.) converges absolutely and uniformly for 0, z Λ a,δ. Proof. We first prove by induction that r n, z) ) n ) ) c a + Imz) na + Imz)... e na, n. n! z a a + 2 Imz) na + 2 Imz) According to 2.2) we have r 0, z) = and, from 2.2) and 3.), r, z) c e at + e ta+2 Imz))+2 Imz)) dt, 2 z which yields r, z) c ) a + Imz) e a. a z a + 2 Imz) The result is therefore true for n =. Suppose that it were true for k n, n 2. According to 2.2) we have r n, z) qt)r n t, z) + e 2t ) Imz)) dt, 2 z so that, from 3.) and the induction hypothesis, ) n ) c c a + Imz) r n, z)... 2 z n )! z a a + 2 Imz) ) n )a + Imz) + e nat + e 2t ) Imz) )dt, n )a + 2 Imz) which yields r n, z) ) n ) ) ) c a + Imz) n )a + Imz) na + Imz)... e na, n! z a a + 2 Imz) n )a + 2 Imz) na + 2 Imz)
EJDE-2005/45 ZEROS OF THE JOST FUNCTION 5 as required. The lemma is proved when we notice that 0 < if Imz) > a/3 and z > δ > /a. na + Imz) na + 2 Imz) < 2, n, and z a < < We are now in position to identify a region in the z-plane where χz) cannot vanish. Theorem 3.2. Suppose 3.) holds and fi δ > /a. Then, for z Λ a,δ, ) χz) 2 ep In particular, if δ > a ln2), then χz) cannot vanish inside the set Λ a,δ and the operator L 0 has i) no eigenvalue λ = z 2 such that z Λ a,δ {z : Imz) > 0}, ii) no spectral singularity λ = z 2 such that z, δ) δ, + ), iii) no resonance inside Λ a,δ {z : Imz) < 0}. Proof. According to lemma 3. we have, for z Λ a,δ, r n, z) ) n e na, 0, n! so that Since we obtain r n, z) n! ) n ) e na = ep e a. χ, z) = e Imz) + r n, z) e Imz){ } r n, z), In particular, χz) does not vanish if i.e. if χz) 2 ep 2 ep from which i), ii) and iii) follow. δ > ). ) > 0, a ln2), Note that, under the hypotheses of theorem 3.2, if λ = z 2 is an eigenvalue of L 0 then z can only be located on the semi disk {z C : z δ, Imz) > 0} and, if q is real-valued, on the segment line z = it, 0 < t δ. Also, under the hypotheses of theorem 3.2, the resonances situated on {z C : a/3 < Imz) < 0} must be inside the set {z C : a/3 < Imz) < 0, z δ} and the spectral singularities λ = z 2 must satisfy δ < z < δ. We now show that a similar situation prevails in the case α 0.
6 D. GILBERT, A. KEROUANTON EJDE-2005/45 Lemma 3.3. Suppose that 3.) holds and fi δ > /a. Then s n, z) z ) ne na, 0, Imz) > a/3, n n! z a and the series 2.3) converges absolutely and uniformly for 0, z Λ a,δ. Proof. From 2.2), 2.3) and 2.4), we have d d χ, z) = eiz iz + ) s n, z) and s n, z) z 2 z qt)r n t, z) + e 2 Imz)t )) dt, n. Arguing as in lemma 3., we obtain the stated result. The bounds we obtain for α 0, π/2) π/2, π) are not as tight as the ones obtained in theorem 3.2, which is rather natural as, for α 0, π/2) π/2, π), it is possible to find resonances far below the real ais or large eigenvalues, depending on the value of α. We refer to the first eample in the net section for an illustration of this phenomenon. Theorem 3.4. Suppose that 3.) holds and let δ be such that δ > a ln2). Then i), ii) and iii) of theorem 3.2 hold as they stand for the operator L π/2 and i), ii) and iii) of theorem 3.2 continue to hold for the operator L α, α 0, π/2) π/2, π), provided we replace δ by ma{δ, δ α }, where δ α = cotα) ) ep 2 ep ). Proof. We first suppose that α = π/2. According to.5), 2.3) and lemma 3.3 we have, for z Λ a,δ, { χ π/2 z) z z ep ) } { = z 2 ep ) }. 3.2) It follows that χ π/2 z) cannot vanish inside Λ a,δ if δ > /a ln2), and the first part of the theorem is proved. Suppose now that α 0, π/2) π/2, π). From 2.) and lemma 3. we get χz) + On the other hand, according to.5), so that, with 3.2), we obtain r n 0, z) ep ). χ α z) sinα)χ π/2 z) cosα)χz) χ α z) z sinα) { 2 ep )} ) cosα) ep.
EJDE-2005/45 ZEROS OF THE JOST FUNCTION 7 From the equality above, it is not hard to see that χ α z) > 0 for ep ) z > cotα) 2 ep ), from which the last part of the theorem follows. Let δ = ma {δ, δ α }. Under the hypotheses of theorem 3.4, the eigenvalues λ = z 2 must be such that z {z C : z δ }, the resonances situated on {z C : a/3 < Imz) < 0} must be inside the set {z C : a/3 < Imz) < 0, z δ } and the spectral singularities λ = z 2 must satisfy δ < z < δ. 4. Eamples The case q 0. Let q 0 in.). Then the Jost solution is χ, z) = e iz so that χ α, z) = cosα)e iz + iz sinα)e iz, α 0, π). Hence the only zero of χ α z) is z = 0 if α = π/2 z α = i cotα) if α 0, π/2) π/2, π). If α 0, π/2) then Imz α ) > 0, so that λ α = cot 2 α) is an eigenvalue and, if α π/2, π), then Imz α ) < 0 so that z α = i cotα) is a resonance. If we suppose that α is strictly comple then sinh2 Imα)) z α = cosh2 Imα)) cos2 Reα)) + i sin2 Reα)) cosh2 Imα)) cos2 Reα)), so that λ α = zα 2 is an eigenvalue if sin2 Reα)) > 0, and z α is a resonance if sin2 Reα)) < 0 and λ α = zα 2 is a spectral singularity if sin2 Reα)) = 0. The Jost-Bessel function. If we take q) = be d in.), with b, d C and Red) > 0, then it can be proved by induction [7] that, in the notation of 2.), χ, z) = e iz{ + } r n, z) = e iz{ + bd 2 e d ) n ) n! 2iz/d)... }. n 2iz/d) This formula for the Jost solution is independently confirmed in [2], where it is noted that when q is real valued,.) is satisfied by the Bessel function J 2iz/d {2id b)e d/2}, which is in L 2 R + ) for Imz) > 0 see also [3, 4.4] and [4, 2.3]). If d > 0 and b > 0, then as in [2] L 0 had no eigenvalues and also no antibound states in the segment line z = it, d/2 < t < 0. Taking b = and d =, it was shown in [7], using methods we have not discussed in the present paper, that although L 0 has no eigenvalues, it does have a unique antibound state z 0 = it 0 such that t 0 /2, 0), more precisely t 0 [ 0.39, 0.2]. In order to compare the last of these eamples with the results obtained in theorem 3.2, take a = and c = in theorem 3.2. Theorem 3.2 predicts that if δ 2.9, then
8 D. GILBERT, A. KEROUANTON EJDE-2005/45 χz) has no zero inside the set {z C : Imz) > /3, z > δ}, so that the estimate obtained in the last eample is consistent with the bound obtained in theorem 3.2. Note that the bounds obtained in theorem 3.2 with a = and c = also apply, for eample, to the comple valued potential q) = i + i e +2i). 5. Jost function and Titchmarsh-Weyl function We suppose in the first instance that q LR + ) is real valued and give a brief account of the relationship between the Jost function and the Titchmarsh-Weyl function, since the eigenvalues and more generally the spectrum of the operator L α have traditionally been studied using the properties of the Titchmarsh-Weyl function m α λ). Let φ α, λ) be defined as above and let θ α, λ) be the solution of.) satisfying θ α 0, λ) = cosα), θ α0, λ) = sinα). Since Weyl s limit-point case applies at +, it is known that there eists a unique linearly independent L 2 R + )-solution ψ α of.) such that ψ α, λ) = θ α, λ) + m α φ α, λ), 0, Im λ > 0, which is known as the Weyl solution [3]. The function m α λ) is analytic in the upper half plane {λ C : Imλ) > 0} and satisfies Im m α λ)) > 0 for Imλ) > 0, so that lim Im λ 0+ m α λ) eists and is finite Lebesgue almost everywhere. eigenvalues of L α are the poles of m α. On the other hand, it is readily seen that The χ, z) = W 0 χ, φ α )θ α, z 2 ) + W 0 θ α, χ)φ α, z 2 ), Imz) > 0, so that we have formally It follows that ψ α, z 2 ) = χ, z). W 0 χ, φ α ) m α z 2 ) = W 0θ α, χ) W 0 χ, φ α ) = W 0θ α, χ), Imz) > 0, Rez) > 0 5.) χ α z) and the poles of m α z 2 ) are the zeros of χ α z). Since W 0 θ α, χ) and χ α z) are analytic in the upper half plane {z C : Imz) > 0}, we can analytically etend m α λ) using 5.). The etended Titchmarsh-Weyl function is meromorphic on C \ [0, + ). If q LR + ) is allowed to be comple valued and if Imq) 0, a similar situation prevails [2] and we can construct a Titchmarsh-Weyl function which is analytic on {λ C : Imλ) > 0} and can be analytically etended to a function meromorphic on C \ [0, + ). For additional information and references on the relationship between the Jost solution and the Titchmarsh-Weyl function, we refer to [6].
EJDE-2005/45 ZEROS OF THE JOST FUNCTION 9 References [] M. S. P. Eastham, The Asymptotic Solution of Linear Differential Systems, Applications of the Levinson Theorem, London Mathematical Society Monographs New Series, Clarendon Press, Oford, 989. [2] M. S. P. Eastham, Antibound states and eponentially decaying Sturm-Liouville potentials, J. London Math. Soc. 2) 65 2002) 624-638. [3] G. Freiling, V. Yurko, Inverse Sturm-Liouville Problems and their Applications, Nova Science Publishers, New York, 200. [4] R. Froese, Asymptotic distribution of resonances in one dimension, Journal of Differential Equations 37 997) 25-272. [5] M. Hitrik, Bounds on scattering poles in one dimension, Comm. Math. Phys. 208 999) 38-4. [6] S. V. Hruščev, Spectral singularities of dissipative Schrödinger operators with rapidly decaying potentials, Indiana University Math. Journal Vol. 33 No. 4 984) 63-638. [7] A. Kerouanton, Self- and nonself-adjoint Sturm-Liouville operators with eponentially decaying potentials, PhD thesis, Dublin Institute of Technology 2005). [8] K. Kodaira, The eigenvalue problem for ordinary differential equations of the second order and Heisenberg s theory of S-matrices, Amer. J. Math. 7 949) 92-945. [9] M. A. Naimark, Investigation of the spectrum and the epansion in eigenfunctions of a nonselfadjoint differential operator of the second order on a semi-ais, American Mathematical Translation Series 2 Volume 6 966). [0] M. A. Naimark, Linear Differential Operators, Part II, Harrap, London, 968. [] A. Rybkin, Some new and old asymptotic representations of the Jost solution and the Weyl m-function for Schrödinger operators on the line, Bull. London Math. Soc. 34 2002) 6-72. [2] A. R. Sims, Secondary conditions for linear differential operators of the second order, Journal of Mathematics and Mechanics, Vol. 6, No. 2 957). [3] E. C. Titchmarsh, Eigenfunction Epansions, Part I, Second Edition, Clarendon, Oford, 962. [4] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 962. Daphne Gilbert School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland E-mail address: daphne.gilbert@dit.ie Alain Kerouanton School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland E-mail address: alainkerouanton@hotmail.com Fa: +3534024994