On the exact solutions of a one-dimensional Schr\"odinger equation with a rational potential

TL;DR

This paper investigates exact solutions of a one-dimensional Schrödinger equation with a rational potential, establishing a link between special eigenvalues and the true spectrum through eigenfunction node analysis.

Contribution

It introduces a method to connect isolated exact eigenvalues with the actual spectrum using eigenfunction nodes in a conditionally-solvable model.

Findings

Derived a relation between exact and actual eigenvalues

Identified conditions for eigenfunction nodes

Provided insights into the spectral properties of rational potentials

Abstract

We analyse the exact solutions of a conditionally-solvable Schr\"odinger equation with a rational potential. From the nodes of the exact eigenfunctions we derive a connection between the otherwise isolated exact eigenvalues and the actual eigenvalues of the Hamiltonian operator.