Intertwining operators for non self-adjoint Hamiltonians and bicoherent states

Abstract

This paper is devoted to the construction of what we will call {\em exactly solvable models}, i.e. of quantum mechanical systems described by an Hamiltonian $H$ whose eigenvalues and eigenvectors can be explicitly constructed out of some {\em minimal ingredients}. In particular, motivated by PT-quantum mechanics, we will not insist on any self-adjointness feature of the Hamiltonians considered in our construction. We also introduce the so-called bicoherent states, we analyze some of their properties and we show how they can be used for quantizing a system. Some examples, both in finite and in infinite-dimensional Hilbert spaces, are discussed.

arXiv commentIn press in Journal of Mathematical Physics

References in corpus (22)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Quantization Methods: A Guide for Physicists and Analysts
• Operator (quasi-)similarity, quasi-Hermitian operators and all that
• Two families of superintegrable and isospectral potentials in two dimensions
• Intertwined isospectral potentials in an arbitrary dimension
• Intertwined Hamiltonians in Two Dimensional Curved Spaces
• Intertwining Operator Realization of Non-Relativistic Holography
• Position Dependent Mass Schroedinger Equation and Isospectral Potentials : Intertwining Operator approach
• A non self-adjoint model on a two dimensional noncommutative space with unbound metric
• Mathematical aspects of intertwining operators: the role of Riesz bases
• Model pseudofermionic systems: connections with exceptional points
• Intertwining operators between different Hilbert spaces: connection with frames
• Non linear pseudo-bosons
• Non linear pseudo-bosons versus hidden Hermiticity. II: The case of unbounded operators
• $\mathcal{D}$-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization
• Non linear pseudo-bosons versus hidden Hermiticity
• Investigation of PT-symmetric Hamiltonian systems from an alternative point of view
• Non Hermitian Operators with Real Spectrum in Quantum Mechanics
• Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation
• $\mathcal{D}$ pseudo-bosons in quantum models
• Non self-adjoint Hamiltonians with complex eigenvalues

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• A description of pseudo-bosons in terms of nilpotent Lie algebras
• Generalized Heisenberg algebra and (non linear) pseudo-bosons
• Coordinate representation for non Hermitian position and momentum operators
• A concise review on pseudo-bosons, pseudo-fermions and their relatives
• Non-Hermitian coherent states for finite-dimensional systems
• Some Consequences of the Grunewald-O'Halloran Conjecture for Pseudoquonic Operators