The number of Dirac-weighted eigenvalues of Sturm-Liouville equations
with integrable potentials and an application to inverse problems
Abstract
In this paper, we further Meirong Zhang, et al.’s work by computing the
number of weighted eigenvalues for Sturm-Liouville equations, equipped
with general integrable potentials and Dirac weights, under Dirichlet
boundary condition. We show that, for a Sturm-Liouville equation with a
general integrable potential, if its weight is a positive linear
combination of $n$ Dirac Delta functions, then it has at most $n$
(may be less than $n$, or even be $0$) distinct real Dirichlet
eigenvalues, or every complex number is a Dirichlet eigenvalue; in
particular, under some sharp condition, the number of Dirichlet
eigenvalues is exactly $n$. Our main method is to introduce the
concepts of characteristics matrix and characteristics polynomial for
Sturm-Liouville problem with Dirac weights, and put forward a general
and direct algorithm used for computing eigenvalues. As an application,
a class of inverse Dirichelt problems for Sturm-Liouville equations
involving single Dirac distribution weights is studied.