On the Spectrum of Eigenparameter-Dependent Quantum Difference Equations

Transcription

1 Appl. Math. Inf. Sci. 9, No. 4, Applied Mathematics & Information Sciences An International Journal On the Spectrum of Eigenparameter-Dependent Quantum Difference Euations Yelda Aygar Martin J. Bohner 2, Department of Mathematics, Faculty of Sciences, University of Ankara, 6, Ankara, Turkey 2 Missouri S&T, Department of Mathematics Statistics, 4 West 2th Street, Rolla, MO , USA Received: Oct. 24, Revised: Jan. 25, Accepted: Jan. 25 Published online: Jul. 25 Abstract: We consider a boundary value problem BVP consisting of a second-order uantum difference euation boundary conditions depending on an eigenvalue parameter. Discussing the point spectrum using the uniueness theorem of analytic functions, we present a condition that guarantees that this BVP has a finite number of eigenvalues spectral singularities with finite multiplicities. Keywords: Quantum difference euation, discrete spectrum, spectral analysis, spectral singularities, eigenvalues Introduction Consider the BVP consisting of the Sturm Liouville euation y + Qxy=λ 2 y, x< y hy=, where Q is a complex-valued function, h C, λ is a spectral parameter. Spectral analysis of BVP was investigated by Naĭmark 2]. He showed that the spectrum of BVP is composed of the eigenvalues, the continuous spectrum the spectral singularities. The spectral singularities are poles of the resolvent s kernel which are imbedded in the continuous spectrum are not eigenvalues. Boundary value problems for difference euations have been intensively studied in last decade in order to investigate problems in engineering, economics control theory. Spectral theory of difference euations has been investigated by some authors in connection with the classical moment problem 4, 5, ]. Some problems of spectral theory for difference euations were also treated in, 6 9]. Furthermore, spectral analysis of -difference euations with spectral singularities has been investigated in 2, 3]. In this paper, we let > use the notation N := n : n N, where N denotes the set of nonnegative integers. Let us consider the nonselfadjoint BVP consisting of the second-order -difference euation t t atyt + btyt + a y = λ yt, t N 2 the boundary conditions + λy+β + β λy=, β β, β a, 3 whereat t N bt t N are complex seuences, λ is a spectral parameter, at for all t N, i,β i C, i=,. The set up of this paper is summarized as follows: Section 2 discusses the Jost solution Jost function of the BVP 2 3. Also, we give the Green function resolvent of this BVP in this section. In Section 3, we investigate the eigenvalues the spectral singularities of the BVP 2 3 get some properties of the eigenvalues the spectral singularities of this BVP under the condition sup t N exp ε lnt ] at + bt <, ε >, 2. 4 Corresponding author bohner@mst.edu c 25 NSP

2 726 Y. Aygar, M. Bohner: On the Spectrum of Eigenparameter-Dependent... In Section 4, we deal with the condition 4 for =. For both cases, we prove that the BVP 2 3 has a finite number of eigenvalues spectral singularities with finite multiplicities. Since the second case is weaker than the first, we have to use a different way for each to prove the theorem. is the resolvent of the BVP 2 3, where l 2 N is the Hilbert space of complex-valued functions with the inner product f,g := t N µt ftgt, f,g : N C. 2 Jost Solution Jost Function Assume 4. Then 2 has the solution lnt ei et,z=αt z µt + r N At,re i z, t N, for λ = 2 cosz, where αt, At,r are expressed in terms of at bt, z C + :=z C:Imz µt = t for all t N 2]. Moreover, At,r satisfies At,r C s t 2, 5 N as + bs, 6 where 2 is the integer part of 2 C > is a constant. Therefore, e, z is analytic with respect to z in C + :=z C:Imz> continuous inc +. Using 5 the boundary condition 3, we define the function f by fz = + 2 cosze,z +β β cosze,z. The function f is analytic in C +, continuous in C +, fz = fz+2π. Analogously to the Sturm Liouville differential euation, the solution e, z the function f are called the Jost solution Jost function of 2 3, respectively 3]. Let ϕλ = ϕt,z t N, be the solution of 2 satisfying the initial conditions ϕ,λ= + λ, ϕ,λ=β + β λ. If we define φt,z=ϕ2 cosz=ϕt,2 cosz t N, then φ is an entire function φz = φz+2π. Let us define the semi-strips P =z C + : 2 π Rez 3π 2 P=P 2. For all z P with fz, we define the Green function of the BVP 2 3 by φr,zet,z a fz, r= t k, k N G t,z z := er,zφt,z a fz, r= 8 tk, k N. It is obvious that Rht := r N Gt,rhr, h l 2 N 9 3 Eigenvalues Spectral Singularities We will denote the set of all eigenvalues spectral singularities of BVP 2 3 by σ d σ ss, respectively. Using 8, 9, the definition of the eigenvalues the spectral singularities 3], we get σ d =λ C:λ = 2 cosz, z P, fz=, σ ss = λ C:λ = 2 cosz, z π 2, 3π 2 ], fz= \. From 5 7, we find fz = α β e iz β + α + α + α + α β e iz If we define +α e 2iz + α β A,re i r N + α A,r + r N β +α A,r α A,r z e i z +α β A,r e i z + + α A,re i +2 z. Fz := fze iz, 2 then we get Fz = α β β + α + α e iz c 25 NSP

3 Appl. Math. Inf. Sci. 9, No. 4, / α + α β +α e 3iz + α β A,re i z r N + α A,r β +α A,r + α A,r r N e 2iz e i + z +α β A,r e i z +2 + α A,re i +3 z. 3 Since f is analytic in C +, continuous in C + fz = fz+2π, the function F is also analytic in C +, continuous in C +, Fz = Fz+2π. It follows from 2 that σ d =λ C:λ = 2 cosz, z P, Fz=, 4 σ ss = λ C:λ = 2 cosz, z π 2, 3π 2 ], Fz= \. 5 Definition. The multiplicity of a zero of F in P is called the multiplicity of the corresponding eigenvalue or spectral singularity of BVP 2 3. Using 4 5, we get that in order to investigate the uantitative properties of the BVP 2 3, we need to discuss the uantitative properties of the zeros of F in P. Let us define M := z P : Fz=, M 2 := z 6 2 : Fz=. We also denote the set of all limit points of M by M 3 the set of all zeros of F with infinite multiplicity in P by M 4. From 4 6, we get that σ d = λ C:λ = 2 cosz, z M, σ ss = λ C:λ = 2 cosz, z M 2 \. 7 Theorem. Assume 4. Then i the set M is bounded countable, ii M M 3 = /, M M 4 = /, iii the set M 2 is compact the Lebesgue measure of M 2 in the real axis is zero, iv M 3 M 2, M 4 M 2, the Lebesgue measure of M 3 M 4 are also zero, v M 3 M 4. Proof. Using 6 3, for all z P, we find Fz= β α+oe Imz, β, Imz, Fz= α+β α]e iz + Oe 2Imz, β =, Imz. These euations show the boundedness of the set M. Since F is a 2π-periodic function is analytic in C +, we get that M has at most a countable number of elements. ii iv can be obtained from the boundary uniueness theorem of analytic functions ]. We can easily get v using the continuity of all derivatives of F on π 2, 3π ] 2. Now we can give the following theorem as a result of Theorem 7. Theorem 2. Assume 4. Then the set σ d is bounded, has at most countable number of elements its limit points can lie only in 2,2 ]. Also σ ss 2,2 ] the Lebesgue measure of the set σ ss in the real axis is zero. 4 Main Result Let us suppose that the complex seuencesat t N bt t N satisfy exp ε lnt at + bt <, ε >. sup t N It is clear that 4 reduces to 8 for =. 8 Theorem 3. Assume 8. Then the BVP 2 3 has a finite number of eigenvalues spectral singularities, each of them is of finite multiplicity. Proof. It follows from 6 8 that At,r C exp ε, t,, r N. 9 4 By using 9, we observe that the function F has an analytic continuation to the half-plane Imz> ε 4. So, the limit points of its zeros in P cannot lie in 2. From Theorem, we get that the bounded sets M M 2 have no limit points, i.e., the sets M M 2 have a finite number of elements. Using the analyticity of F in Imz > ε 4, we find that all zeros of F in P have finite multiplicity. Conseuently, we get the finiteness of the eigenvalues the spectral singularities of the BVP 2 3. c 25 NSP

4 728 Y. Aygar, M. Bohner: On the Spectrum of Eigenparameter-Dependent... In the following, we will assume that ] exp at + bt sup t N ε lnt <, ε >, 2 <, 2 which is weaker than 8. As is known, the condition 8 guarantees the analytic continuation of F from the real axis to the lower half-plane. So, we get the finiteness of the eigenvalues the spectral singularities of BVP 2 3 as a result of this analytic continuation. It follows from 2 that the function F is analytic in C + infinitely differentiable on the real axis. But F does not have an analytic continuation from the real axis to the lower half-plane. Therefore, under the condition 2, the finiteness of the eigenvalues the spectral singularities of the BVP 2 3 cannot be proved by the same techniue used in Theorem 3. We will use the following uniueness theorem 8, Lemma 4.4] for analytic functions in order to prove the next theorem. Theorem 4. Assume that the 2π-periodic function g is analytic inc +, all of its derivatives are continuous inc +, sup g k z η k, k N. z P If the set G 2 with Lebesgue measure zero is the set of all zeros of the function g with infinity multiplicity in P, if w lntsdµg s =, where ts = inf k N η k s k k! µg s is the Lebesgue measure of the s-neighborhood of G, w > is an arbitrary constant, then g inc +. Lemma. Assume 2. Then the ineuality holds, where F k z η k, z P, k N 2 η k C4 k + Dd k k!k k, 22 D d are positive constants depending on C, ε. Proof. Using 6 2, we obtain At,r C exp ε, t,, r N It follows from 3 23 that where F k z C4 k + D k, z P, k N, D k = C4 k r N k e 4 ε, k N. We can also write for D k D k = C4 k m= m k e ε 4 m n = C4 k t k e 4 ε t dt C4 k t k e 4 ε t dt. If we define y= ε 4 t, then we get k+ 4 D k C4 k ε using the Gamma function, we obtain D k C4 2k+ 4 ε k+ Using 24 the ineualities k+ < e k, k k < k!e k, we have y k+ e y dy, k+ k+ k. 24 D k Dd k k!k k, k N, + k k < e, where D d are positive constants depending on ε. Lemma 2. If 2 holds, then M 4 = /. Proof. Since the function F is not eual to zero, we can write w lntsdµm 4,s> 25 η by using Theorem 4, where ts = inf k s k k N k!, µm 4,s is the Lebesgue measure of s-neighborhood of M 4, η k is defined by 22. Substituting 22 in the definition of ts, we find ts=dexp It follows from that w e ds s dµm 4,s<.. 26 The last ineuality holds for arbitrary s if only if µm 4,s=, i.e., M 4 = /. This completes the proof. Theorem 5. Under the condition 2, the BVP 2 3 has a finite number of eigenvalues spectral singularities, each of them is of finite multiplicity. Proof. To be able to prove this, we have to show that the function F has a finite number of zeros with finite multiplicities in P. Using Theorem Lemma 2, we obtain that M 3 = /. So the bounded sets M M 2 have no limit points, i.e., the function F has only finite number of zeros in P. Since M 4 = /, these zeros are of finite multiplicity. c 25 NSP

5 Appl. Math. Inf. Sci. 9, No. 4, / References ] Murat Adıvar Elgiz Bairamov. Spectral properties of non-selfadjoint difference operators. J. Math. Anal. Appl., 262:46 478, 2. 2] Murat Adıvar Martin Bohner. Spectral analysis of -difference euations with spectral singularities. Math. Comput. Modelling, 437-8:695 73, 26. 3] Murat Adıvar Martin Bohner. Spectrum principal vectors of second order -difference euations. Indian J. Math., 48:7 33, 26. 4] Ravi P. Agarwal. Difference euations ineualities, volume 228 of Monographs Textbooks in Pure Applied Mathematics. Marcel Dekker Inc., New York, second edition, 2. Theory, methods, applications. 5] N. I. Akhiezer. The classical moment problem some related uestions in analysis. Translated by N. Kemmer. Hafner Publishing Co., New York, ] Yelda Aygar Elgiz Bairamov. Jost solution the spectral properties of the matrix-valued difference operators. Appl. Math. Comput., 289: , 22. 7] Yelda Aygar, Murat Olgun, Turhan Koprubasi. Principal functions of nonselfadjoint discrete Dirac euations with spectral parameter in boundary conditions. Abstr. Appl. Anal., pages Art. ID , 5, 22. 8] Elgiz Bairamov, Öner Çakar, Allan M. Krall. Nonselfadjoint difference operators Jacobi matrices with spectral singularities. Math. Nachr., 229:5 4, 2. 9] Elgiz Bairamov Cafer Coskun. The structure of the spectrum of a system of difference euations. Appl. Math. Lett., 84: , 25. ] E. P. Dolzhenko. Boundary-value uniueness theorems for analytic functions. 256: , 979. ] Walter G. Kelley Allan C. Peterson. Difference euations. Harcourt/Academic Press, San Diego, CA, second edition, 2. An introduction with applications. 2] M. A. Naĭmark. Investigation of the spectrum the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis. Amer. Math. Soc. Transl. 2, 6:3 93, 96. 3] M. A. Naĭmark. Linear differential operators. Part II: Linear differential operators in Hilbert space. With additional material by the author, a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt. Frederick Ungar Publishing Co., New York, 968. Yelda Aygar is a Research Assistant at the Department of Mathematics at Ankara University in Ankara, Turkey. She received the BS 26, MS 28, PhD 23 in Mathematics from Ankara University. She spent one year as a Postdoctoral Researcher funded by the Turkish Council of Higher Education, YÖK in the area of uantum difference euations at the Department of Mathematics Statistics at Missouri University of Science Technology in Rolla, Missouri, USA. Her research interests are spectral analysis, functional analysis, difference euations operators, uantum difference euations. She also is a recipient of the governmental undergraduate student scholarship Turkey, a Turkish Science Research Council Graduate Student Fellowship TÜBITAK, Martin Bohner is the Curators Professor of Mathematics Statistics at Missouri University of Science Technology in Rolla, Missouri, USA. He received the BS 989 MS 993 in Econo-mathematics PhD 995 from Universität Ulm, Germany, MS 992 in Applied Mathematics from San Diego State University. His research interests center around differential, difference, dynamic euations as well as their applications to economics, finance, biology, physics, engineering. He is the author of four textbooks more than 2 publications, Editor-in-Chief of two international journals, Associate Editor for more than 5 international journals, President of ISDE, the International Society of Difference Euations. Professor Bohner s honors at Missouri S&T include five Faculty Excellence Awards, one Faculty Research Award, eight Teaching Awards. c 25 NSP