Reconstruction of potential and boundary conditions for second order difference equations

• Autorzy: Currie S. , Love A.

Identyfikatory

DOI

10.14708/cm.v58i1-2.5133

• Języki publikacji: EN

Abstrakty

EN

Assume the eigenvalues and the weights are given for a difference boundary value problem and that the form of the boundary conditions at the endpoints is known. In particular, it is known whether the endpoints are fixed (i.e. Dirichlet or non-Dirichlet boundary conditions) or whether the endpoints are free to move (i.e. boundary conditions with affine dependence on the eigenparameter). This work illustrates how the potential as well as the exact boundary conditions can be uniquely reconstructed. The procedure is inductive on the number of unit intervals. This paper follows along the lines of S. Currie and A. Love, Inverse problems for difference equations with quadratic eigenparameter dependent boundary conditions, Quaestiones Mathematicae, 40 (2017), no. 7, 861-877. Since the inverse problem considered in this paper contains more unknowns than the inverse problem considered in the above reference, an additional spectrum is required more often than was the case in the unique reconstruction of the potential alone.

Słowa kluczowe

EN

difference equations; inverse problem; boundary value problems; potential; eigenvalues

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2018
• Tom: Vol. 58, No. 1/2
• Strony: 1--9
• Opis fizyczny: Bibliogr. 7 poz., tab.

Twórcy

autor

Currie S.

• School of Mathematics, University of theWitwatersrand, Private Bag 3, PO WITS 2050, South Africa

autor

Love A.

• School of Mathematics, University of theWitwatersrand, Private Bag 3, PO WITS 2050, South Africa

Bibliografia

• [1] M. Bohner and H. Koyunbakan, Inverse problems for Sturm-Liouville difference equations, Filomat 30 (2016), no. 5, 1297-1304, DOI 10.2298/FIL1605297B.
• [2] S. Currie and A. Love, Inverse problems for difference equations with quadratic eigenparameter dependent boundary conditions, Quaest. Math. 40 (2017), no. 7, 861-877, DOI 10.2989/16073606.2017.1334156.
• [3] S. Currie and A. Love, Hierarchies of difference boundary value problems continued, Adv. Differ. Equ. 19 (2013), no. 11, 1807-1827, DOI 10.1186/1687-1847-2013-311.
• [4] S. Elaydi, An introduction to difference equations, Springer 2005.
• [5] C. Gao and R. Ma, Eigenvalues of discrete Sturm-Liouville problems with eigenparameter dependent boundary conditions, Linear Algebra Appl. 503 (2016), 100-119, DOI 10.1016/j.laa.2016.03.043.
• [6] O. H. Hald, Discrete inverse Sturm-Liouville problems. I. Uniqueness for symmetric potentials, Numer. Math. 27 (1976/77), no. 2, 249-256.
• [7] W. G. Kelley and A. C. Peterson, Difference equations: an introduction with applications, Academic Press, New York 1991.