An existence criterion for the $\mathcal{PT}$-symmetric phase transition

Emanuela Caliceti; Sandro Graffi

doi:10.3934/dcdsb.2014.19.1955

Article Contents

2014, Volume 19, Issue 7: 1955-1967. Doi: 10.3934/dcdsb.2014.19.1955

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An existence criterion for the $\mathcal{PT}$-symmetric phase transition

Emanuela Caliceti^1, and
Sandro Graffi^1,

1.
Dipartimento di Matematica, Università di Bologna, 40127 Bologna

Received: March 31, 2013

Revised: June 30, 2013

Published: August 2014

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Abstract

We consider on $L^2(\mathbb{R})$ the Schrödinger operator family $H(g)$ with domain and action defined as follows $$ D(H(g))=H^2(\mathbb{R})\cap L^2_{2M}(\mathbb{R}); \quad H(g) u=\bigg(-\frac{d^2}{dx^2}+\frac{x^{2M}}{2M}-g\,\frac{x^{M-1}}{M-1}\bigg)u $$ where $g\in\mathbb{C}$, $M=2,4,\ldots\;$. $H(g)$ is self-adjoint if $g\in\mathbb{R}$, while $H(ig)$ is $\mathcal{PT}$-symmetric. We prove that $H(ig)$ exhibits the so-called $\mathcal{PT}$-symmetric phase transition. Namely, for each eigenvalue $E_n(ig)$ of $H(ig)$, $g\in\mathbb{R}$, there exist $R_1(n)>R(n)>0$ such that $E_n(ig)\in\mathbb{R}$ for $|g| < R(n)$ and turns into a pair of complex conjugate eigenvalues for $|g| > R_1(n)$.

Keywords:

$PT$-symmetry,
phase transitions.

Mathematics Subject Classification: 34Lxx, 81Q10, 81Q12, 81Q15.

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