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Journal of Physics A Mathematical and Theoretical

An analytic family of {\cal P}{\cal T} -symmetric Hamiltonians with real eigenvalues

Emanuela Caliceti, F. Cannata, S. Graffi

Published Jun 3, 2008• 6 citations
Publisher

Abstract

We study nonlinear perturbative expansions for -symmetric local Schrödinger operators. The Schrödinger operator is a sum of the harmonic oscillator Hamiltonian and a local -symmetric potential depending, in general, nonlinearly on the perturbation parameter. A specific class of models having real spectrum for any value of the parameter is proposed.

DOI
10.1088/1751-8113/41/24/244008

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We study nonlinear perturbative expansions for -symmetric local Schrödinger operators. The Schrödinger operator is a sum of the harmonic oscillator Hamiltonian and a local -symmetric potential depending, in general, nonlinearly on the perturbation parameter. A specific class of models having real spectrum for any value of the parameter is proposed.
We discuss three Hamiltonians, each with a central-field part H0H_{0}H0​ and a PT-symmetric perturbation igzigzigz. When H0H_{0}H0​ is the isotropic Harmonic oscillator the spectrum is real for all ggg because HHH is isospectral to H0+g2/2H_{0}+g^{2}/2H0​+g2/2. When H0H_{0}H0​ is the Hydrogen atom then infinitely many eigenvalues are complex for all ggg. If the potential in H0H_{0}H0​ is linear in the radial variable rrr then the spectrum of HHH exhibits real eigenvalues for $0
We study a class of PT-symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in \cite{BoMe14} that, when the potential is analytic, the eigenvalues stay real for a perturbation of size O(1). We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.
We consider a two-parameter non-Hermitian quantum mechanical Hamiltonian operator that is invariant under the combined effects of parity and time reversal transformations. Numerical investigation shows that for some values of the potential parameters the Hamiltonian operator supports real eigenvalues and localized eigenfunctions. In contrast with other parity times time reversal symmetric models which require special integration paths in the complex plane, our model is integrable along a line parallel to the real axis.
A major mathematical problem in PT-symmetric quantum mechanics is to determine whether or not the spectrum of any given non self-adjoint but PT-symmetric Schroedinger operator is real. In this case the PT-symmetry is called "proper". Clearly in this connection an equally important issue is the spontaneous violation of the PT-symmetry which might occur in a PT-symmetric operator family. The spontaneous violation of the PT-symmetry is defined as the transition from real values of the spectrum to complex ones at the variation of the parameter labeling the family. Its occurrence is referred to also as the PTPTPT-symmetric phase transition. \nThis chapter intends to present a review of recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators. The main technical instrument is represented by perturbation theory.
Negative-energy PT\mathcal {P}\mathcal {T}PT-symmetric Hamiltonians
The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian H=p2+x2(ix)εH=p^2+x^2(ix)^εH=p2+x2(ix)ε has real, positive, and discrete eigenvalues for all ε≥0ε\geq 0ε≥0. These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues En=2n+1E_n=2n+1En​=2n+1 (n=0, 1, 2, 3, ...) at ε=0ε=0ε=0. However, the harmonic oscillator also has negative eigenvalues En=−2n−1E_n=-2n-1En​=−2n−1 (n=0, 1, 2, 3, ...), and one may ask whether it is equally possible to continue analytically from these eigenvalues. It is shown in this paper that for appropriate PT-symmetric boundary conditions the Hamiltonian H=p2+x2(ix)εH=p^2+x^2(ix)^εH=p2+x2(ix)ε also has real and {\it negative} discrete eigenvalues. The negative eigenvalues fall into classes labeled by the integer N (N=1, 2, 3, ...). For the Nth class of eigenvalues, εεε lies in the range $(4N-6)/3