Harmonic Oscillator with a Step and its Isospectral Properties

Abstract

We investigate the one-dimensional Schrödinger equation for a harmonic oscillator with a finite jump $a$ at the origin. The solution is constructed by employing the ordinary matching-of-wavefunctions technique. For the special choices of $a$, $a=4\ell$ ($\ell =1,2,\dots$), the wavefunctions can be expressed by the Hermite polynomials. Moreover, we explore isospectral deformations of the potential via the Darboux transformation. In this context, infinitely many isospectral Hamiltonians to the ordinary harmonic oscillator are obtained.

18 pages, 8 figures; Published version

References in corpus (3)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator

Cited by in corpus (4)

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