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Harmonic Oscillator with a Step and its Isospectral Properties

Yuta Nasuda, Nobuyuki Sawado

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 $a$, $a=4\ell$ ($\ell=1,2,\ldots$), 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.

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 at the origin. The solution is constructed by employing the ordinary matching-of-wavefunctions technique. For the special choices of , (), 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.
Paper Structure (13 sections, 44 equations, 8 figures, 2 tables)

This paper contains 13 sections, 44 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. The matching of wavefunctions at the origin
  4. The Solutions
  5. Case for arbitrary $a$
  6. Hermite-polynomial solutions for $a=4\ell$ ($\ell = 1,2,\ldots$)
  7. On the algebraic equation \ref{['eq:algebraic']}
  8. Isospectral Hamiltonians
  9. Crum's theorem
  10. Isospectrality to 1-d harmonic oscillator
  11. Further deformation: Krein--Adler transformation
  12. Conclusion
  13. Explicit forms of the negative energy eigenvalues for $V(x;24)$

Figures (8)

  • Figure 1: The potential \ref{['eq:pot']}. Here, the parameter $a$ is set to be $4$ as an example.
  • Figure 2: Graphical solution of Eq. \ref{['eq:E-det_a_transcental']}. The blue curves correspond to the left hand side of the equation, while the red ones are the right hand side. The intersections of these curves determine the energy eigenvalues. The numerical solutions are displayed in Tab. \ref{['tab:ene_2']}.
  • Figure 3: The solutions of the eigenvalue problem \ref{['eq:SEa']} with $a=2$. Thin blue lines show the energy spectrum, and the blue curve on each line is the corresponding eigenfunction. The potential $V(x;2)$ is also plotted in this figure by a black curve.
  • Figure 4: The solutions of the eigenvalue problem \ref{['eq:SE4l']} with $\ell=1$. The potential $V(x;4)$ is displayed in this figure by a black curve. Thin lines show the energy spectrum, and the colored curve on each line is the corresponding eigenfunction. The states plotted in yellow possess the Hermite-polynomial solvability, while that colored in blue does not.
  • Figure 5: The solutions of the eigenvalue problem \ref{['eq:SE4l']} with $\ell=6$. The potential $V(x;24)$ is displayed in this figure by a black curve. Thin lines show the energy spectrum, and the colored curve on each line is the corresponding eigenfunction. The states plotted in yellow possess the Hermite-polynomial solvability, while those colored in blue do not.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5