Exactly solvable piecewise analytic double well potential $V_{D}(x)=min[(x+d)^2,(x-d)^2]$ and its dual single well potential $V_{S}(x)=max[(x+d)^2,(x-d)^2]$

Ryu Sasaki

doi:10.1063/5.0127371

paper

Exactly solvable piecewise analytic double well potential $V_{D} (x) = min [(x + d)^{2}, (x - d)^{2}]$ and its dual single well potential $V_{S} (x) = ma x [(x + d)^{2}, (x - d)^{2}]$

arXiv:2209.09445 · doi:10.1063/5.0127371

Abstract

By putting two harmonic oscillator potential $x^{2}$ side by side with a separation $2 d$ , two exactly solvable piecewise analytic quantum systems with a free parameter $d > 0$ are obtained. Due to the mirror symmetry, their eigenvalues $E$ for the even and odd parity sectors are determined exactly as the zeros of certain combinations of the confluent hypergeometric function $_{1} F_{1}$ of $d$ and $E$ , which are common to $V_{D}$ and $V_{S}$ but in two different branches. The eigenfunctions are the piecewise square integrable combinations of $_{1} F_{1}$ , the so called $U$ functions. By comparing the eigenvalues and eigenfunctions for various values of the separation $d$ , vivid pictures unfold showing the tunneling effects between the two wells.

arXiv commentA few typos corrected

Exactly solvable piecewise analytic double well potential $V_{D} (x) = min [(x + d)^{2}, (x - d)^{2}]$ and its dual single well potential $V_{S} (x) = ma x [(x + d)^{2}, (x - d)^{2}]$

Abstract

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Exactly solvable piecewise analytic double well potential VD​(x)=min[(x+d)2,(x−d)2] and its dual single well potential VS​(x)=max[(x+d)2,(x−d)2]

Abstract

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Exactly solvable piecewise analytic double well potential $V_{D} (x) = min [(x + d)^{2}, (x - d)^{2}]$ and its dual single well potential $V_{S} (x) = ma x [(x + d)^{2}, (x - d)^{2}]$