Non-Hermitian Quantum Mechanics of Non-diagonalizable Hamiltonians: puzzles with self-orthogonal states

Abstract

We consider QM with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells in particular biorthogonal bases. The "self-orthogonality" phenomenon is clarified in terms of a correct spectral decomposition and it is shown that "self-orthogonal" states never jeopardize resolution of identity and thereby quantum averages of observables. The example of a complex potential leading to one Jordan cell in the Hamiltonian is constructed and its origin from level coalescence is elucidated. Some puzzles with zero-binorm bound states in continuous spectrum are unraveled with the help of a correct resolution of identity.

25 pages, one subsection and an appendix added, terminology is refined, important refs.added

References in corpus (7)

• Introduction to PT-Symmetric Quantum Theory
• Higher-order supersymmetric quantum mechanics
• Nonlinear Supersymmetry for Spectral Design in Quantum Mechanics
• Biorthogonal Quantum Systems
• Application of Pseudo-Hermitian Quantum Mechanics to a PT-Symmetric Hamiltonian with a Continuum of Scattering States
• Is the CPT-norm always positive?
• How to Test for Diagonalizability: The Discretized PT-Invariant Square-Well Potential

Cited by in corpus (34)

• Pseudo-Hermitian Representation of Quantum Mechanics
• Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at real Energies
• Symmetries and conserved quantities in Lindblad master equations
• Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart
• Projective Hilbert space structures at exceptional points
• PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain
• Nonlinear Supersymmetric Quantum Mechanics: concepts and realizations
• New features of scattering from a one-dimensional non-Hermitian (complex) potential
• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties
• New supersymmetry-generated complex potentials with real spectra
• Optical lattices with exceptional points in the continuum
• Bi-Orthogonal Approach to Non-Hermitian Hamiltonians with the Oscillator Spectrum: Generalized Coherent States for Nonlinear Algebras
• Effective Hamiltonian approach to adiabatic approximation in open systems
• Spectral singularities for Non-Hermitian one-dimensional Hamiltonians: puzzles with resolution of identity
• Gamow-Siegert functions and Darboux-deformed short range potentials
• Quadratic PT-symmetric operators with real spectrum and similarity to self-adjoint operators
• Interplay between Riccati, Ermakov and Schroedinger equations to produce complex-valued potentials with real energy spectrum
• Spectral singularity and deep multiple minima in the reflectivity in non-Hermitian (complex) Ginocchio potential
• Effective Hamiltonian Approach to Open Systems and Its Applications
• PT symmetry breaking and exceptional points for a class of inhomogeneous complex potentials
• Exactly Solvable Non-Separable and Non-Diagonalizable 2-Dim Model with Quadratic Complex Interaction
• Fraudulent White Noise: Flat power spectra belie arbitrarily complex processes
• Hermitian Hamiltonian equivalent to a given non-Hermitian one. Manifestation of spectral singularity
• Pseudo-Hermiticity of an Exactly Solvable Two-Dimensional Model
• Optical potentials using resonance states in Supersymmetric Quantum Mechanics
• Non-adiabatic transitions in a non-symmetric optical lattice
• Reflectionless PT-symmetric potentials in the one-dimensional Dirac equation
• Equidistance of the Complex 2-Dim Anharmonic Oscillator Spectrum: Exact Solution
• Extended WKB method, resonances and supersymmetric radial barriers
• Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
• Antiperiodic XXZ chains with arbitrary spins: Complete eigenstate construction by functional equations in separation of variables
• Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs
• Three-Dimensional Shape Invariant Non-Separable Model With Equidistant Spectrum
• Quantum backreaction effect in optical solitons