Algebraic and spectral properties of some quasiorthogonal polynomials encountered in quantum radiation

Abstract

The nodal structure of the wavefunctions of a large class of quantum-mechanical potentials is often governed by the distribution of zeros of real quasiorthogonal polynomials. It is known that these polynomials (i) may be described by an arbitrary linear combination of two orthogonal polynomials {Pn(x)} and (ii) have real and simple zeros. Here, the three term recurrence relation, the second order differential equation and the distribution of zeros of quasiorthogonal polynomials of the classical class (i.e., when Pn(x) is a Jacobi, Laguerre or Hermite polynomial) are derived and analyzed. Specifically, the exact values of the Newton sum rules and the WKB density of zeros of these polynomials are found.