Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians

Abstract

For a large class of time-dependent non-Hermitain Hamiltonians expressed in terms linear and bilinear combinations of the generators for an Euclidean Lie-algebra respecting different types of PT-symmetries, we find explicit solutions to the time-dependent Dyson equation. A specific Hermitian model with explicit time-dependence is analyzed further and shown to be quasi-exactly solvable. Technically we constructed the Lewis-Riesenfeld invariants making useof the metric picture, which is an equivalent alternative to the Schrödinger, Heisenberg and interaction picture containing the time-dependence in the metric operator that relates the time-dependent Hermitian Hamiltonian to a static non-Hermitian Hamiltonian.

arXiv comment14 pages

References in corpus (25)

• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermitian Representation of Quantum Mechanics
• PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold
• Time-dependent version of cryptohermitian quantum theory
• Unitary quantum evolution for time-dependent quasi-Hermitian systems with non-observable Hamiltonians
• Time-Dependent Pseudo-Hermitian Hamiltonians Defining a Unitary Quantum System and Uniqueness of the Metric Operator
• Time evolution of non-Hermitian Hamiltonian systems
• Operator (quasi-)similarity, quasi-Hermitian operators and all that
• Time Dependent $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics
• A note on the PT-invariant periodic potential V(x)=4 cos^2 x + 4 i V_0 sin 2x
• On the invariant method for the time-dependent non-Hermitian Hamiltonians
• Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians
• Energy Observable for a Quantum System with a Dynamical Hilbert Space and a Global Geometric Extension of Quantum Theory
• Mending the broken PT-regime via an explicit time-dependent Dyson map
• Invisible defects in complex crystals
• Solvable two dimensional time-dependent non-Hermitian quantum systems with infinite dimensional Hilbert space in the broken PT-regime
• A Quasi-Exactly Solvable N-Body Problem with the sl(N+1) Algebraic Structure
• Use of Equivalent Hermitian Hamiltonian for $PT$-Symmetric Sinusoidal Optical Lattices
• Metric versus observable operator representation, higher spin models
• E2-quasi-exact solvability for non-Hermitian models
• Spontaneous PT-symmetry breaking for systems of noncommutative Euclidean Lie algebraic type
• Non-Hermitian systems of Euclidean Lie algebraic type with real eigenvalue spectra
• A new non-Hermitian E2-quasi-exactly solvable model
• Analytic solutions for a three-level system in a time-dependent field
• On time-dependent quasi-exactly solvable models

Cited by in corpus (8)

• Time-dependent Darboux (supersymmetric) transformations for non-Hermitian quantum systems
• An introduction to PT-symmetric quantum mechanics -- time-dependent systems
• Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators
• Time-independent approximations for time-dependent optical potentials
• Pseudo-invariant approach for a particle in a complex time-dependent linear potential
• Attoscience in Phase Space
• Time-dependent C-operators as Lewis-Riesenfeld invariants in non-Hermitian theories
• Quantisations of exactly solvable ghostly models