Some Spectral Properties of One Sturm-Liouville Type Problem with Discontinuous Weight

Abstract

We consider a discontinuous weight Sturm-Liouville equation together with eigenparameter dependent boundary conditions and two supplementary transmission conditions at the point of discontinuity. We extend and generalize some approaches and results of the classic regular Sturm-Liouville problems to the similar problems with discontinuities. In particular, we introduce a special Hilbert space formulation in such a way that the problem under consideration can be interpreted as an eigenvalue problem for a suitable selfadjoint operator, construct the Green's function and resolvent operator, and derive asymptotic formulas for eigenvalues and normalized eigenfunctions.

On the resolvent of singular Sturm‐Liouville operators with transmission conditions

“…The regular Sturm‐Liouville problems with transmission conditions have been investigated in the recent papers . However, there are a few studies for the singular case (see previous studies ).…”

Dissipative Sturm-Liouville Operators with Transmission Conditions

“…All the maximal dissipative extensions L of the operator 0 are described by the following conditions (see [4,15,18]):…”

“…Spectral theory is one of the main branches of modern functional analysis and it has many applications in mathematics and applied sciences. There has recently been great interest in spectral analysis of Sturm-Liouville boundary value problems with eigenparameter-dependent boundary conditions (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). Furthermore, many researchers have studied some boundary value problems that may have discontinuities in the solution or its derivative at an interior point [15][16][17][18][19].…”

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

“…The theory of regular Sturm-Liouville problems is well built; since the foundation work of Weyl on limit-point/limit-circle classification [1], the singular Sturm-Liouville problems (see [2][3][4][5][6][7] for real coefficients and [8] for complex coefficients) and more general Hamiltonian systems (see [9,10]) are widely researched. Meanwhile, a large number of researchers are interested in the discontinuous Sturm-Liouville problem with inner discontinuous points, since these problems are of wide applications in engineering and mechanics (see [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]). Various physics applications of this kind of problems are found, such as oscillation of linear or nonlinear equation (see [26][27][28][29]) and heat and mass transfer problems (see [30]).…”