A Singular Sturm-Liouville Problem with Limit Circle Endpoints and Eigenparameter Dependent Boundary Conditions

Abstract

In this paper, we investigate a class of discontinuous singular Sturm-Liouville problems with limit circle endpoints and eigenparameter dependent boundary conditions. Operator formulation is constructed and asymptotic formulas for eigenvalues and fundamental solutions are given. Moreover, the completeness of eigenfunctions is discussed.

Solvability and coerciveness of multi-point Sturm–Liouville problems with abstract linear functionals

“…Note that an abstract theory of the boundary value problems with continuous coefficients and an eigenvalue parameter in the boundary conditions have been constructed by Yakubov and Yakubov (see [4] and corresponding bibliography). Many authors have been devoted to the study of discontinuous problems [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. To deal with the discontinuity of the problem, transmission conditions are imposed on the discontinuous points.…”

Matrix representations of Sturm‐Liouville problems with coupled eigenparameter‐dependent boundary conditions and transmission conditions

“…As we all know, Sturm-Liouville problem with interior discontinuous points inside an interval has been one of the important research topics in mathematical physics, such as heat and mass transfer, vibrating string problems when the string loads with additional point masses, diffraction problems, and so on (see previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and references cited therein). It is well-known that the spectrum of the standard self-adjoint Sturm-Liouville problem is infinite (see Zettl 16 ), and lots of researchers studied the properties of self-adjoint or non-self-adjoint differential operator, such as the deficiency indices, the dependence of eigenvalues, inverse problems, the self-adjoint realization, the oscillation of eigenfunctions, and so on (see other works 8,14,15,[17][18][19][20].…”

“…As an application of other studies, [22][23][24][25][26][27] the researchers studied the inverse Sturm-Liouville problems in two works. 28,29 Meanwhile, a large number of Sturm-Liouville problems with eigenparameter-dependent boundary conditions have also received much attention in research since these problems are applied to physics, engineering, and electric circuits (see previous works 6,[30][31][32][33] ). Such problems should be traced back to Feller who first considered these problems and solved the proper connection in probability theory (see other works 34,35 ).…”