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Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions

Miloslav Znojil

Abstract

Non-stationary version of unitary quantum mechanics formulated in non-Hermitian (or, more precisely, in hiddenly Hermitian) interaction-picture representation is illustrated via an elementary $N$ by $N$ matrix Hamiltonian $H(t)$ mimicking a 1D-box system with physics controlled by time-dependent boundary conditions. The model is presented as analytically solvable at $N=2$. Expressis verbis, this means that for both of the underlying Heisenbergian and Schrödingerian evolution equations the generators (i.e., in our notation, the respective operators $Σ(t)$ and $G(t)$) become available in closed form. Our key message is that contrary to the conventional beliefs and in spite of the unitarity of the evolution of the system, neither its "Heisenbergian Hamiltonian" $Σ(t)$ nor its "Schrödingerian Hamiltonian" $G(t)$ possesses a real spectrum or the conjugate pairs of complex eigenvalues. This means that neither one of these "Hamiltonians" can be pseudo-Hermitian alias PT-symmetric.

Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions

Abstract

Non-stationary version of unitary quantum mechanics formulated in non-Hermitian (or, more precisely, in hiddenly Hermitian) interaction-picture representation is illustrated via an elementary by matrix Hamiltonian mimicking a 1D-box system with physics controlled by time-dependent boundary conditions. The model is presented as analytically solvable at . Expressis verbis, this means that for both of the underlying Heisenbergian and Schrödingerian evolution equations the generators (i.e., in our notation, the respective operators and ) become available in closed form. Our key message is that contrary to the conventional beliefs and in spite of the unitarity of the evolution of the system, neither its "Heisenbergian Hamiltonian" nor its "Schrödingerian Hamiltonian" possesses a real spectrum or the conjugate pairs of complex eigenvalues. This means that neither one of these "Hamiltonians" can be pseudo-Hermitian alias PT-symmetric.
Paper Structure (8 sections, 19 equations, 1 figure)

This paper contains 8 sections, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hiddenly Hermitian quantum mechanics in nuce
  3. Stationary cases and NSP physical Hilbert spaces
  4. Non-stationary NIP and physical Hilbert space
  5. Boundary-controlled square-well model
  6. Robin boundary conditions
  7. Condition number one: the reality of spectrum
  8. Numerics and reparametrizations

Figures (1)

  • Figure :