Solvable non-Hermitian discrete square well with closed-form physical inner product

Abstract

A non-Hermitian $N-$level quantum model with two free real parameters is proposed in which the bound-state energies are given as roots of an elementary trigonometric expression and in which they are, in a physical domain of parameters, all real. The wave function components are expressed as closed-form superpositions of two Chebyshev polynomials. In any eligible physical Hilbert space of finite dimension $N<\infty$ our model is constructed as unitary with respect to an underlying Hilbert-space metric $\Theta \ne I$. The simplest version of the latter metric is finally constructed, at any dimension $N=2,3,\dots$, in closed form. This version of the model may be perceived as an exactly solvable $N-$site lattice analogue of the $N=\infty$ square well with complex Robin-type boundary conditions. At any $N<\infty$ our closed-form metric becomes trivial (i.e., equal to the most common Dirac's metric ${\Theta }^{(Dirac)}=I$) at the special, Hermitian-Hamiltonian-limit parameters.

23 pp., 8 figures

References in corpus (15)

• Making Sense of Non-Hermitian Hamiltonians
• Exponentially Fragile PT-Symmetry in Lattices with Localized Eigenmodes
• PT-Symmetry in Non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials
• Time evolution of non-Hermitian Hamiltonian systems
• Closed formula for the metric in the Hilbert space of a PT-symmetric model
• PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain
• Gegenbauer-solvable quantum chain model
• Maximal couplings in PT-symmetric chain-models with the real spectrum of energies
• Non-Hermitian oscillator Hamiltonian and su(1,1): a way towards generalizations
• Fragile PT-symmetry in a solvable model
• Isospectral Hamiltonians from Moyal products
• Differential Realization of Pseudo-Hermiticity: A quantum mechanical analog of Einstein's field equation
• Discrete quantum square well of the first kind
• Gauging non-Hermitian Hamiltonians
• Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators

Cited by in corpus (5)

• E2-quasi-exact solvability for non-Hermitian models
• Confluences of exceptional points and a systematic classification of quantum catastrophes
• Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions
• Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions
• Emergence and localization of exceptional points in an exactly solvable toy model