Matrix Representations of Eigenparameter-Dependent Dirac Operators With Finite Spectrum

Abstract

In this paper, the finite spectrum problem of Dirac operators with an eigenparameter contained in the boundary conditions is studied. First, the existence of finite spectrum under certain conditions is proved, that is, for each nonnegative integer $m$, we construct a class of regular Dirac operators, each of which has at most $n=2m+2$ eigenvalues. Second, we identify a class of Dirac equations such that every Dirac operator formed by such an equation together with eigenparameter-dependent boundary conditions can be reformulated as an equivalent finite-dimensional matrix eigenvalue problem. Third, for any matrix eigenvalue problem of a prescribed type paired with eigenparameter-dependent boundary conditions, we construct a family of Dirac operators equipped with the specified boundary conditions, each of which is equivalent to the given matrix eigenvalue problem. Here, equivalence signifies that the two problems share identical sets of eigenvalues.