Bound states emerging from below the continuum in a solvable PT-symmetric discrete Schroedinger equation

Abstract

At the lower edge of the energy continuum the birth of an isolated quantum bound state is studied as caused by an infinitesimally small change of the interaction. In our model a single, asymptotically free massive quantum particle is assumed moving along a discretized real line of coordinates, $x\in \mathbb{Z}$. The motion is assumed controlled by a weakly nonlocal 2J-parametric external potential which is non-Hermitian but PT-symmetric. Mathematically, the bound states are then reinterpreted as Sturmians, i.e., the bound-state energy is treated as a variable real parameter while the value of one of the couplings (responsible for the existence of the bound state) is determined via the standard secular equation. It is found that in such an arrangement the model is exactly solvable at all of the finite counts J of the couplings. For illustration, the explicit closed bound-state formulae are presented up to J=7.

arXiv comment29 pages, 3 figures

References in corpus (55)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Supersymmetry and Quantum Mechanics
• Dynamically encircling exceptional points in a waveguide: asymmetric mode switching from the breakdown of adiabaticity
• Pseudo-Hermitian Representation of Quantum Mechanics
• Pump-induced Exceptional Points in Lasers
• Physical realization of $\mathcal{P}\mathcal{T}$-symmetric potential scattering in a planar slab waveguide
• Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures
• PT -symmetric harmonic oscillators
• A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points
• General description of quasi-adiabatic dynamical phenomena near exceptional points
• Observation of a Chiral State in a Microwave Cavity
• Generalized Continuity Equation and Modified Normalization in PT-Symmetric Quantum Mechanics
• Pseudospectra in non-Hermitian quantum mechanics
• Robust and fragile PT-symmetric phases in a tight-binding chain
• On the metric operator for the imaginary cubic oscillator
• Operator (quasi-)similarity, quasi-Hermitian operators and all that
• Scattering in PT-symmetric quantum mechanics
• Closed formula for the metric in the Hilbert space of a PT-symmetric model
• Statistics of resonance width shifts as a signature of eigenfunction non-orthogonality
• Conceptual Problems in Scattering from Localized non-Hermitian Potentials
• Breaking of PT-symmetry in bounded and unbounded scattering systems
• PT-symmetry breaking in the steady state of microscopic gain-loss systems
• Resonances in open quantum systems
• Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation
• Scattering theory with localized non-Hermiticities
• Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime
• Maximal couplings in PT-symmetric chain-models with the real spectrum of energies
• Experimental Width Shift Distribution: A Test of Nonorthogonality for Local and Global Perturbations
• Discrete PT-symmetric models of scattering
• Pseudo-Hermitian Quantum Mechanics with Unbounded Metric Operators
• Solvable PT-symmetric model with a tunable interspersion of non-merging levels
• Conditional observability
• Quantum catastrophes: a case study
• Complete Set of Inner Products for a Discrete PT-symmetric Square-well Hamiltonian
• The problem of coexistence of several non-Hermitian observables in PT-symmetric quantum mechanics
• Scaling behavior and phase diagram of a PT-symmetric non-Hermitian Bose-Hubbard system
• An exactly solvable quantum-lattice model with a tunable degree of nonlocality
• Matching method and exact solvability of discrete PT-symmetric square wells
• Non-Hermitian spectral effects in a PT-symmetric waveguide
• Multiply Degenerate Exceptional Points and Quantum Phase Transitions
• Matrix Hamiltonians with an algebraic guarantee of unbroken PT-symmetry
• Three solvable matrix models of a quantum catastrophe
• Non linear pseudo-bosons versus hidden Hermiticity. II: The case of unbounded operators
• Thermodynamics of Quantum Heat Bath
• Competing PT potentials and re-entrant PT symmetric phase for a particle in a box
• Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding
• Quantum inner-product metrics via recurrent solution of Dieudonne equation
• Hilbert space inner products for PT-symmetric Su-Schrieffer-Heeger models
• Negative-energy PT-symmetric Hamiltonians
• Sturm-Schroedinger equations: formula for metric
• Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions
• On S-matrix of Schrodinger Operators with Non-Symmetric Zero-Range Potentials
• PT-symmetric model with an interplay between kinematical and dynamical non-localities
• Eigenvalues collision for PT-symmetric waveguide

Cited by in corpus (5)

• Bell non-locality and Kochen-Specker contextuality: How are they connected?
• Simple models of three coupled $\mathcal{P}\mathcal{T}$-symmetric wave guides allowing for third-order exceptional points
• PT-symmetric dynamical confinement: Fermi acceleration, quantum force and Berry phase
• Analytic approximation for eigenvalues of a class of $\mathcal{P}\mathcal{T}$ symmetric Hamiltonians
• Resonances and continued-fraction Green's functions in non-Hermitian Bose-Hubbard-like quantum models