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

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

URL: http://arxiv.org/abs/2209.09445v1
Date: Tue, 20 Sep 2022 03:46:03 GMT
Title: Exactly solvable piecewise analytic double well potential $V_{D}(x)=min[(x+d)^2,(x-d)^2]$ and its dual single well potential $V_{S}(x)=max[(x+d)^2,(x-d)^2]$
Authors: Ryu Sasaki
Abstract summary: Two piecewise analytic quantum systems with a free parameter $d>0$ are obtained. Their eigenvalues $E$ for the even and odd parity sectors are determined. vivid pictures unfold showing the tunneling effects between the two wells.
Score: 0.0
License: http://creativecommons.org/licenses/by/4.0/
Abstract: By putting two harmonic oscillator potential $x^2$ side by side with a separation $2d$, 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 ${}_1F_1$ 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 ${}_1F_1$, 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.

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