International Journal of Analysis and Applications Volume 16, Number 1 (2018), 25-37 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-25 EQUIVALENCE OF STURM-LIOUVILLE PROBLEM WITH FINITELY MANY δ-INTERACTIONS AND MATRIX EIGENVALUE PROBLEMS ABDULLAH KABLAN∗, MEHMET AKİF ÇETİN Faculty of Arts and Sciences, Department of Mathematics, Gaziantep University, Gaziantep, 27310, Turkey ∗Corresponding author: kablan@gantep.edu.tr Abstract. The purpuse of this article is to show the matrix representations of Sturm-Liouville operators with finitely many δ-interactions. We show that a Sturm-Liouville problem with finitely many δ-interactions can be represented as a finite dimensional matrix eigenvalue problem which has the same eigenvalue with the former Sturm-Liouville operator. Moreover an example is also presented. 1. Introduction Acording to classical spectral theory, a Sturm–Liouville problem (SLP) consisting of the equation −(py′)′ + qy = λwy, on J = (a,b) and boundary conditions has infinite spectrum under some assumptions. Atkinson in his book [1] suggested that if the coefficients of SLP satisfy some conditions, the problem may have finite eigenvalues. Then in [2], Kong, Wu and Zettl obtained the following result: For every positive integer n, we can construct a class of regular self-adjoint and nonself-adjoint SLP with exactly n eigenvalues by choosing p and w such that 1/p and w are alternatively zero on consecutive subintervals. Recently, there has been much attention paid to the SLPs with finite spectrum. For a comprehensive treatment of the subject we refer the reader to the book by Zettl [3], and the papers by Kong, Wu and Zettl [2], Ao, Sun, and Zhang [4], [5] and Ao, Bo and Sun [6], [7]. In 2009, the equivalence of SLP with Received 7th September, 2017; accepted 20th October, 2017; published 3rd January, 2018. 2010 Mathematics Subject Classification. 34B24, 47A10, 15A18. Key words and phrases. one-dimensional Schrödinger operator; finite spectrum; point interactions. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 25 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-25 Int. J. Anal. Appl. 16 (1) (2018) 26 a matrix eigenvalue problem was first constructed by Volkmer and Zettl in [8]. By equivalance of matrix eigenvalue problems for the SLPs with finite spectrum we mean to construct a matrix eigenvalue problem with exactly the same eigenvalues as the corresponding SLP. Then, the matrix representations of SLPs with finite spectrum are extended to various problems. For the SLPs see [8]- [11] and for fourth order boundary value problems see [12]- [16]. The goal of this paper is to find the matrix representation of the following Sturm-Liouville problem with finitely many δ-interactions: − (py′)′ + ∞∑ n=1 αnδ(x−xn)y + qy = λwy, on J = (a,b), (1.1) where J = (a,x1) ∪ (x1,x2) ∪ ...∪ (xn,b), x1, ...,xn ∈ (a,b) with −∞ < a < b < ∞, αj’s are real numbers, δ(x) is the Dirac delta function and λ ∈ C is a spectral parameter. Sturm-Liouville equations with Dirac delta function potentials often appear in quantum mechanics. For example, such an equations had been used for modelling of atomic and molecular systems including atomic lattices, quantum heterostructures, semiconductors, organic fluorescent materials, solar cells etc. (see [17], [18], [19] and citations of them). Recently, we generalize the finite spectrum result to the problem (1.1) in [20]. The equation (1.1) is equivalent to the many-point boundary value problem, (see [19]). So we can understand problem (1.1) as studying the equation − (py′)′ + qy = λwy, on J, (1.2) and n transmission conditions CjY (xj−) = Y (xj+), Y =   y py′   , j = 1, 2, ...,n (1.3) where xj’s are inner discontinuity points and Cj =   1 0 αj 1   . Additionally, let us consider the boundary conditions of the form AY (a) + BY (b) = 0, A,B ∈ M2(C) (1.4) where A = (aij)2×2, B = (bij)2×2 are complex valued 2 × 2 matrices and M2(C) denotes the set of square matrices of order 2 over C. Here, the coefficients fulfill the following minimal conditions: r = 1 p , q, w ∈ L(J, C), (1.5) where L(J, C) denotes the complex valued functions which are Lebesgue integrable on J. Int. J. Anal. Appl. 16 (1) (2018) 27 The BC (1.3) is said to be self-adjoint if the following two conditions are satisfied: rank(A,B) = 2, AEA∗ = BEB∗ with E =  0 −1 1 0   . (1.6) It is well known that under the condition (1.5), the BCs (1.3) fall into two disjoint classes: seperated and coupled. The seperated boundary conditions have the canonical representation: cos αy(a) − sin α(py′)(a) = 0, 0 ≤ α < π (1.7) cos βy(b) − sin β(py′)(b) = 0, 0 < β ≤ π. The real coupled boundary conditions have the canonical representation: Y (b) = KY (a) with K = (ks,t)2×2, ks,t ∈ R, det(K) = 1. (1.8) Let u = y and v = (py′). Then we have the system representation of equation (1.2) u′ = rv, v′ = (q −λw)u, on J. (1.9) 2. Matrix representations of SLPs with Finitely Many δ-Interactions Definition 2.1. A Sturm-Liouville equation with finitely many δ-interactions (1.1) or equivalently the equa- tion (1.2) with transmission condition (1.3) is said to be of Atkinson type if, for some integers mj ≥ 1, j = 0, 1, ...,n, there exists a partition of the interval J a = x00 < x01 < x02 < ... < x0,2m0+1 = x1, (2.1) x1 = x10 < x11 < x12 < ... < x1,2m1+1 = x2, ... xn−1 = xn−1,0 < xn−1,1 < xn−1,2 < ... < xn−1,2mn−1+1 = xn, xn = xn0 < xn1 < xn2 < ... < xn,2mn+1 = b such that for each j ∈{0, 1, ...,n} r = 1 p = 0 on (xj,2k; xj,2k+1] , k = 0, 1, ...,mj − 1 and [ xj,2mj ; xj,2mj+1 ) , xj,2k+1∫ xj,2k w 6= 0, xj,2k+1∫ xj,2k q 6= 0, k = 0, 1, ...,mj, (2.2) and q = w = 0 on [xj,2k+1; xj,2k+2] , xj,2k+2∫ xj,2k+1 r 6= 0, k = 0, 1, ...,mj − 1. (2.3) Int. J. Anal. Appl. 16 (1) (2018) 28 Our main aim in this section is to constract matrix eigenvalue problems in such a way that its eigenvalues are exactly the same as those of the corresponding SLPs with finitely many δ-Interactions of Atkinson type. Definition 2.2. A SLP with finitely many δ-Interactions of Atkinson type is said to be equivalent to a matrix eigenvalue problem if the former has exactly the same eigenvalues as the latter. We begin by stating some additional notation. For each j ∈{0, 1, ...,n} given (2.1)-(2.3), let pjk =   xj,2k∫ xj,2k−1 r   −1 , k = 1, 2, ...,mj; qjk = xj,2k+1∫ xj,2k q, wjk = xj,2k+1∫ xj,2k w, k = 0, 1, ...,mj. (2.4) and let introduce the notation m = n∑ j=0 mj. (2.5) We note from (2.2) and (2.3) that pjk, wjk ∈ R�{0} , and no sign restrictions are imposed on them. From (2.2) and (2.3) we can make the following observation: For any solution u, v of (1.9), u is constant on the intervals where r is identically zero and v is constant on the intervals where both q and w are both identically zero. Let u0k = u(x), x ∈ [x0,2k; x0,2k+1] , k = 0, 1, ...,m0 − 1, (2.6) u0m0 = u(x), x ∈ [x0,2m0 ; x0,2m0+1), uj0 = u(x), x ∈ (xj0; xj1] , j = 1, 2, ...,n ujk = u(x), x ∈ [xj,2k; xj,2k+1] , k = 1, 2, ...,mj − 1, j = 1, 2, ...,n− 1 ujmj = u(x), x ∈ [xj,2mj ; xj,2mj+1), j = 0, 1, ...,n− 1 unk = u(x), x ∈ [xn,2k; xn,2k+1], k = 1, 2, ...,mn vjk = v(x), x ∈ [xj,2k−1; xj,2k), k = 1, 2, ...,mj, j = 0, 1, ...,n and set vj0 = v(xj0+), vj,mj+1 = v(xj,2mj+1−), j = 0, 1, ...,n. (2.7) Lemma 2.1. Assume Eq. (1.2) is of Atkinson type. Then for each j = 0, 1, ...,n and for any solution u, v of Eq. (1.9), we have pjk(ujk −uj,k−1) = vjk, k = 1, 2, ...,mj, (2.8) vj,k+1 −vjk = ujk(qjk −λwjk), k = 0, 1, ...,mj. (2.9) Int. J. Anal. Appl. 16 (1) (2018) 29 Conversely, for any solution ujk, k = 0, 1, ...,mj and vjk, k = 0, 1, ...,mj + 1 of system (2.8), (2.9), there is a unique solution u(x) and v(x) of Eq. (1.9) satisfying (2.6) and (2.7). Proof. Relying on the first equation of (1.9), for k = 1, 2, ...,mj, we have ujk −uj,k−1 = u(xj,2k) −u(xj,2k−2) = xj,2k∫ xj,2k−2 u′ = xj,2k∫ xj,2k−2 rv = xj,2k∫ xj,2k−1 rv = vjk xj,2k∫ xj,2k−1 r = vjk�pjk. This establishes (2.8). Similarly, from second equation of (1.9), for k = 0, 1, ...,mj, we have vj,k+1 −vjk = v(xj,2k+1) −v(xj,2k−1) = xj,2k+1∫ xj,2k−1 v′ = xj,2k+1∫ xj,2k−1 (q −λw)u = xj,2k+1∫ xj,2k (q −λw)u = ujk xj,2k+1∫ xj,2k (q −λw) = ujk(qjk −λwjk), which gives (2.9). On the other hand, if ujk, vjk satisfy (2.8) and (2.9), then we define u(x) and v(x) according to (2.6) and (2.7), and then extend them continuously to the whole interval J as a solution of (1.9) by integrating the equations in (1.9) over subintervals. � First, we consider SLP with transmission condition(1.2)-(1.4) with seperated BC (1.7). Theorem 2.1. Assume α ∈ [0,π) , β ∈ (0,π]. Define an (m + 1) × (m + 1) tridiagonal block matrix Pαβ =   M0 N1 M1 N2 M2 . . . . . . Nn Mn Nn+1 Mn+1   and diagonal matrices Qαβ = diag (q00 sin α, q01, ...,q0,m0−1, q0m0 + q10, q11, ...,qn,mn−1, qnmn sin β) , Wαβ = diag (w00 sin α, w01, ...,w0,m0−1, w0m0 + w10, w11, ...,wn,mn−1, wnmn sin β) . Then SLP with transmission conditions (1.2), (1.3), (1.7) is equivalent to matrix eigenvalue problem (Pαβ + Qαβ) U = λWαβU, (2.10) Int. J. Anal. Appl. 16 (1) (2018) 30 where m is as defined in (2.5), U = [u00,u01, ...,u0m0,u11, ...,u1m1, ...,un1, ...,unmn ] T and the matrices Mj’s and Nj’s are defined as follows: M0 = [ p01 sin α + cos α, −p01 sin α ] , (2.11) for each j = 0, 1, ...,n the mj × 2 matrices Nj+1 =   −pj1 pj1 + pj2 0 −pj2 0 0 ... ... 0 0   , (2.12) for each j = 0, 1, ...,n− 1 the mj ×mj matrices Mj+1 =   −pj2 pj2 + pj3 −pj3 −pj3 pj3 + pj4 −pj4 . . . . . . . . . −pj,mj−1 pj,mj−1 + pj,mj −pj,mj −pj,mj pj,mj + pj+1,1 + αj+1 −pj+1,1   , (2.13) and the mn × (mn − 1) matrix Mn+1 =   −pn2 pn2 + pn3 −pn3 −pn3 pn3 + pn4 −pn4 . . . . . . . . . −pn,mn−1 pn,mn−1 + pn,mn −pn,mn −pn,mn sin β pn,mn sin β − cos β   . (2.14) Proof. For each j = 0, 1, ...,n and k = 1, 2, ...,mj − 1, there is one-to-one correspondence between the solutions of system (2.8), (2.9) and the solutions of the following system: pj1(uj1 −uj0) −vj0 = uj0(qj0 −λwj0), (2.15) pj,k+1(uj,k+1 −ujk) −pjk(ujk −uj,k−1) = ujk(qjk −λwjk), (2.16) vj,mj+1 −pjmj (ujmj −ujmj−1 ) = ujmj (qjmj −λwjmj ) (2.17) Int. J. Anal. Appl. 16 (1) (2018) 31 Therefore, by Lemma 2.1, any solution of equation (1.9), and hence of (1.2), is uniquely determined by a solution of system (2.15)-(2.17). Note that from boundary condition (1.7), we have u00 cos α = v00 sin α (2.18) unmn cos β = vn,mn+1 sin β and for each j = 0, 1, ...,n− 1 from the transmission condition (1.3), we have uj+1,0 = ujmj (2.19) vj+1,0 = αj+1ujmj + vj,mj+1. Additionally, for each j = 0, 1, ...,n− 1 from the equations (2.15)-(2.19), we have pj+1,1 ( uj+1,1 −ujmj ) −pjmj ( ujmj −uj,mj−1 ) −ujmj ( qjmj −λwjmj ) (2.20) = αj+1ujmj + ujmj (qj+1,0 −λwj+1,0) pj+1,2 (uj+1,2 −uj+1,1) −pj+1,1 ( uj+1,1 −ujmj ) = uj+1,1 (qj+1,1 −λwj+1,1) . (2.21) Then the equivalence follows from (2.15)-(2.18) and (2.20), (2.21). � Corollary 2.1. Assume α, β ∈ (0,π) . Define the (m + 1) × (m + 1) tridiagonal block matrix Pαβ =   M0 N1 M1 N2 M2 . . . . . . Nn Mn Nn+1 Mn+1   and diagonal matrices Qαβ = diag (q00,q01, ...,q0,m0−1,q0m0 + q10,q11, ...,qn,mn−1,qnmn ) Wαβ = diag (w00,w01, ...,w0,m0−1,w0m0 + w10,w11, ...,wn,mn−1,wnmn ) . Then SLP with transmission conditions (1.2), (1.3), (1.7) is equivalent to matrix eigenvalue problem (Pαβ + Qαβ) U = λWαβU (2.22) where U = [u00,u01, ...,u0m0,u11, ...,u1m1, ...,un1, ...,unmn ] T , Int. J. Anal. Appl. 16 (1) (2018) 32 and the matrices Mj’s and Nj’s are defined as in Theorem 2.1 except M0 and Mn+1. In this case, M0 is 1 × 2 matrix M0 = [ p01 + cot α −p01 ] , and Mn+1 is mn × (mn − 1) matrix Mn+1 =   −pn2 pn2 + pn3 −pn3 −pn3 pn3 + pn4 −pn4 . . . . . . . . . −pn,mn−1 pn,mn−1 + pn,mn −pn,mn −pn,mn pn,mn − cot β   . Proof. If we divide the first and the last rows of system (2.10) by sin α and sin β respectively, then we obtain (2.22). � Theorem 2.1 and its Corollary show that the problem (1.2)-(1.4), (1.7) of Atkinson type have represen- tations by tridiagonal matrix eigenvalue problems. Now, we will show that the problem (1.2)-(1.4), (1.8) of Atkinson type also have representations. Theorem 2.2. Consider the boundary condition (1.8) with k12 = 0. Define the m × m matrix which is tridiagonal except for the (1,m) and (m, 1) entries P1 =   M0 −k11pnmn N1 M1 N2 M2 . . . . . . Nn Mn −k11pnmn Nn+1 Mn+1   and diagonal matrices Q1 = diag ( q00 + k 2 11qnmn,q01, ...,q0,m0−1,q0m0 + q10,q11, ...,qnmn ) , W1 = diag ( w00 + k 2 11wnmn,w01, ...,w0,m0−1,w0m0 + w10,w11, ...,wnmn ) . Then SLP with transmission conditions (1.2), (1.3), (1.8) is equivalent to matrix eigenvalue problem (P1 + Q1) U = λW1U (2.23) where U = [ u00,u01, ...,u0m0,u11, ...,u1m1, ...,un1, ...,un,mn−1 ]T , Int. J. Anal. Appl. 16 (1) (2018) 33 and the elements of the matrix P1 are defined as follows: The 1 × 2 matrix M0 = [ −k11k21 + p01 + k211pnmn −p01 ] , for each j = 0, 1, ...,n− 1 the mj × 2 and for j = n the (mn − 1) × 2 matrices Nj+1 =   −pj1 pj1 + pj2 0 −pj2 0 0 ... ... 0 0   , for each j = 0, 1, ...,n− 1 the mj ×mj matrices Mj+1 =   −pj2 pj2 + pj3 −pj3 −pj3 pj3 + pj4 −pj4 . . . . . . . . . −pj,mj−1 pj,mj−1 + pj,mj −pj,mj −pj,mj pj,mj + pj+1,1 + αj+1 −pj+1,1   , and the (mn − 1) × (mn − 2) matrix Mn+1 =   −pn2 pn2 + pn3 −pn3 −pn3 pn3 + pn4 −pn4 . . . . . . . . . −pn,mn−2 pn,mn−2 + pn,mn−1 −pn,mn−1 −pn,mn−1 pn,mn−1 + pnmn   . Proof. As mentioned before, the transmission condition (1.3) is the same as (2.19). On the other hand, since k12 = 0, the boundary condition (1.8) is represented as follows: unmn = k11u00 (2.24) un,mn+1 = k21u00 + k22v00 where k11k22 = 1. We find out that for each j = 0, 1, ...,n − 1 and k = 0, 1, ...,mj − 1 there is one-to-one correspondence between the solutions consisting of system (2.8), (2.9), (2.19), (2.24) and the solutions of the Int. J. Anal. Appl. 16 (1) (2018) 34 following system: [ −k11k21 + k211 ( pj+1,mj+1 + qj+1,mj+1 −λwj+1,mj+1 )] uj0 (2.25) = (λwj0 −pj1 −qj0)uj0 + pj1uj1 + k11pj+1,mj+1uj+1,mj+1−1 pj,k+1 (uj,k+1 −ujk) −pjk (ujk −uj,k−1) = ujk (qjk −λwjk) (2.26) pj+1,1 (uj+1,1 −uj+1,0) −vj+1,0 = uj+1,0(qj+1,0 −λwj+1,0) (2.27) pj+1,mj+1 ( k11uj0 −uj+1,mj+1−1 ) −pj+1,mj+1−1uj+1,mj+1−1 (2.28) = pj+1,mj+1−1uj+1,mj+1−2 + uj+1,mj+1−1(qj+1,mj+1−1 −λwj+1,mj+1−1) Then, by Lemma 2.1, any solution of system (1.9), hence of (1.2), is uniquely determined by a solution of system (2.25)-(2.28). � Theorem 2.3. Consider the boundary condition (1.8) with k12 6= 0. Define the (m + 1) × (m + 1) matrix which is tridiagonal except for the (1,m + 1) and (m + 1, 1) entries P2 =   M0 1 k12 N1 M1 N2 M2 . . . . . . Nn Mn 1 k12 Nn+1 Mn+1   and diagonal matrices Q2 = diag (q00,q01, ...,q0,m0−1,q0m0 + q10,q11, ...,qn,mn−1,qnmn ) W2 = diag (w00,w01, ...,w0,m0−1,w0m0 + w10,w11, ...,wn,mn−1,wnmn ) Then SLP with transmission conditions (1.2), (1.3), (1.8) is equivalent to matrix eigenvalue problem (P2 + Q2) U = λW2U (2.29) where U = [u00,u01, ...,u0m0,u11, ...,u1m1, ...,un1, ...,unmn ] T , and the elements of the matrix P3 are defined as follows: For each j = 0, 1, ...,n the matrices Nj+1 ’s and for each j = 0, 1, ...,n−1 the matrices Mj+1 ’s are defined as in Theorem 2.2. On the other hand, the 1×2 matrix M0 = [ p01 − k11k12 −p01 ] , Int. J. Anal. Appl. 16 (1) (2018) 35 and the mn × (mn − 1) matrix Mn+1 =   −pn2 pn2 + pn3 −pn3 −pn3 pn3 + pn4 −pn4 . . . . . . . . . −pn,mn−1 pn,mn−1 + pnmn −pnmn −pnmn pnmn − k22 k12   . Proof. The boundary condition (1.8) can be represented as follows: unmn = k11u00 + k12v00, vn,mn+1 = k21u00 + k22v00. Since k11k22 −k12k21 = 1, we have from the this condition that v00 = − k11 k12 u00 + 1 k12 unmn, vn,mn+1 = − 1 k12 u00 + k22 k12 unmn. On the other hand, if we consider the transmission condition (2.19), the proof is similar with Theorem 2.2. � 3. Example In this section, we give an example to illustrate that a SLP with finitely many δ-interactions and it’s equivalent matrix eigenvalue problem, we will construct it, have same eigenvalues. Consider the SLP with δ-interactions on J = (−3, 0) ∪ (0, 6), − (py′)′ + δ(x− 0)y + qy = λwy. (3.1) This equation is equivalent to the following SLP − (py′)′ + qy = λwy (3.2) with transmission condition   y(0−) −y(0+) = 0y(0−) + py′(0−) −py′(0+) = 0. (3.3) By choosing α = 0 and β = π, we consider the following boundary conditions  y(−3) = 0y(6) = 0. (3.4) Int. J. Anal. Appl. 16 (1) (2018) 36 In this case, the matrices in (1.3) and (1.4) become C1 =   1 0 1 1   , A =   1 0 0 0   , B =   0 0 1 0   respectively. Now, let’s take a partition of the interval J as follows: a = −3 < −2 < −1 < 0 = x1 (3.5) x1 = 0 < 2 < 3 < 4 < 5 < 6 = x2 = b. This yields that m0 = 1, m1 = 2 and define the piecewise constant functions p, q, w are as follows: p(x) =   ∞, (−3,−2) 1, (−2,−1) ∞, (−1, 0) ∞, (0, 2) 1 2 , (2, 3) ∞, (3, 4) 1 4 , (4, 5) ∞, (5, 6) q(x) =   0, (−3,−2) 0, (−2,−1) 1, (−1, 0) 2, (0, 2) 0, (2, 3) 3, (3, 4) 0, (4, 5) 4, (5, 6) w(x) =   1, (−3,−2) 0, (−2,−1) 3, (−1, 0) 4, (0, 2) 0, (2, 3) 1, (3, 4) 0, (4, 5) 2, (5, 6) (3.6) By using the similar method as given in [4], [5] or [20] we have the following two eigenvalues λ1 = 0.67442, λ2 = 3.75739. (3.7) On the other hand, if we find the values pjk, qjk, wjk from (2.4), and use Theorem 2.1 we get the matrices P0π =   1 0 0 0 −1 5 2 −1 2 0 0 −1 2 3 4 −1 4 0 0 0 −1   , Q0π =   0 0 0 0 0 5 0 0 0 0 3 0 0 0 0 0   , W0π =   0 0 0 0 0 11 0 0 0 0 1 0 0 0 0 0   , (3.8) and so the matrix eigenvalue problem (P0π + Q0π) U = λW0πU, (3.9) which is equivalance of SLP with finitely many δ-interactions in (3.1). Indeed, if we find the eigenvalues of the matrix eigenvalue problem (3.9) we obtain the eigenvalues in (3.7). Int. J. Anal. Appl. 16 (1) (2018) 37 References [1] F.V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press-New York, London, 1964. [2] Q. Kong, H. Wu and A. Zettl, Sturm–Liouville Problems with Finite Spectrum, J. Math. Anal. Appl. 263 (2001), 748–762. [3] A. Zettl, Sturm-Liouville Theory, Amer. Math. Soc., Math. Surveys Monographs, no. 121, 2005. [4] J.J. Ao, J. Sun and M.Z. 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