Spectral analysis of a class of symmetric differential operators with logarithmic coefficients

Abstract

Spectral theory of differential operators is a newly emerging interdisciplinary theory in twentieth century. Based on quantum physics, it provides a unified theoretical framework and solution for problems of differential equations. The present paper studies of a class of symmetric differential operators with logarithmic coefficients. Since logarithmic function discrete slower than exponential function and power and integrals of some logarithmic functions cannot be expressed by elementary functions, the paper uses the method of operator decomposition and quadratic comparison. By using several inequalities and convergence of and functions to the corresponding lower bound estimate of quadratic form, it is concluded that under certain conditions, the sufficient condition and necessary condition of discreteness of this kind of differential operators is the spectrum of all self-adjoint extensions.

Keywords:

• Symmetric differential operators
• Discrete spectrum
• Essential spectrum