Spiked harmonic oscillators

TL;DR

This paper provides a comprehensive variational approach to analyze spiked harmonic oscillators, deriving analytical expressions and demonstrating high-accuracy eigenvalue estimations through an optimized basis set.

Contribution

It introduces a complete variational method for spiked harmonic oscillators, including a topological proof of basis completeness and closed-form matrix element expressions.

Findings

Analytical expressions for matrix elements derived.

High-accuracy eigenvalue estimations achieved.

Basis set completeness proven topologically.

Abstract

A complete variational treatment is provided for a family of spiked-harmonic oscillator Hamiltonians H = -d^2/dx^2 + B x^2 + lambda/x^alpha, B > 0, lambda > 0, for arbitrary alpha > 0. A compact topological proof is presented that the set S = {psi_n} of known exact solutions for alpha = 2 constitutes an orthonormal basis for the Hilbert space L_2(0, infinity). Closed-form expressions are derived for the matrix elements of H with respect to S. These analytical results, and the inclusion of a further free parameter, facilitate optimized variational estimation of the eigenvalues of H to high accuracy.