Inverse spectral problems for discontinuous Sturm-Liouville problems of Atkinson type

Abstract We investigate inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type whose spectrum consists of a finite set of eigenvalues. For given two finite sets of interlacing real numbers, there exists a class of Sturm–Liouville equations such that the two sets of numbers are exactly the eigenvalues of their associated Sturm–Liouville problems with two different separated boundary conditions. The main approach is to give an equivalent relation between Sturm–Liouville problems of Atkinson type and matrix eigenvalue problems, and the theory of inverse matrix eigenvalue problems.

[1]  Xiao Wang,et al.  Oscillation criteria for forced second order differential equations with mixed nonlinearities , 2009, Appl. Math. Lett..

[2]  Zhaowen Zheng,et al.  A Singular Sturm-Liouville Problem with Limit Circle Endpoints and Eigenparameter Dependent Boundary Conditions , 2017 .

[3]  Inverse spectral theory for Sturm-Liouville problems with finite spectrum , 2007 .

[4]  Qingkai Kong,et al.  Sturm–Liouville Problems with Finite Spectrum , 2001 .

[5]  Inverse Sturm–Liouville problems with finite spectrum , 2012 .

[6]  Zhaowen Zheng,et al.  Dependence of eigenvalues of 2mth-order spectral problems , 2017 .

[7]  Matrix Representations of Sturm–Liouville Problems with Finite Spectrum , 2009 .

[8]  Ji-jun Ao,et al.  Matrix representations of Sturm-Liouville problems with transmission conditions , 2012, Comput. Math. Appl..

[9]  Jiong Sun,et al.  Spectral analysis for discontinuous non-self-adjoint singular Dirac operators with eigenparameter dependent boundary condition , 2017 .

[10]  Chuan-Fu Yang,et al.  Inverse spectral problems for the Sturm–Liouville operator with discontinuity , 2017, 1703.01403.

[11]  Jiangang Qi,et al.  Classification of Sturm–Liouville differential equations with complex coefficients and operator realizations , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[12]  Zhaowen Zheng,et al.  Invariance of deficiency indices under perturbation for discrete Hamiltonian systems , 2013 .

[13]  William Feller,et al.  The General Diffusion Operator and Positivity Preserving Semi-Groups in One Dimension , 1954 .

[14]  Warren E. Ferguson,et al.  The construction of Jacobi and periodic Jacobi matrices with prescribed spectra , 1980 .

[15]  Ji-jun Ao,et al.  The finite spectrum of Sturm-Liouville problems with transmission conditions , 2011, Appl. Math. Comput..

[16]  Jiong Sun,et al.  Eigenvalues of regular fourth‐order Sturm–Liouville problems with transmission conditions , 2017 .

[17]  Xiao-Ping Yang,et al.  An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions , 2009, Appl. Math. Lett..