$PT$ symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum

Abstract

Consider in $L^2(R^d)$, $d\ge 1$, the operator family $H(g):=H_0+igW$. $\ds H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2$ is the quantum harmonic oscillator with rational frequencies, $W$ a $P$ symmetric bounded potential, and $g$ a real coupling constant. We show that if $|g|<\rho$, $\rho$ being an explicitly determined constant, the spectrum of $H(g)$ is real and discrete. Moreover we show that the operator $\ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1$ has real discrete spectrum but is not diagonalizable.

20 pages

References in corpus (21)

• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• PT-Symmetric Quantum Mechanics
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Must a Hamiltonian be Hermitian?
• Schrödinger operators with complex potential but real spectrum
• Supersymmetry and the spontaneous breakdown of PT symmetry
• On the reality of the eigenvalues for a class of PT-symmetric oscillators
• Supersymmetry without hermiticity
• Large-order Perturbation Theory for a Non-Hermitian PT-symmetric Hamiltonian
• Pseudo-Hermiticity for a Class of Nondiagonalizable Hamiltonians
• Numerical Evidence that the Perturbation Expansion for a Non-Hermitian $\mathcal{P}\mathcal{T}$-Symmetric Hamiltonian is Stieltjes
• Completeness and Orthonormality in PT-symmetric Quantum Systems
• Spectra of PT-Symmetric Operators and Perturbation Theory
• Norm estimates of complex symmetric operators applied to quantum systems
• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties
• On the pseudo-Hermitian nondiagonalizable Hamiltonians
• Two-dimensional SUSY-pseudo-Hermiticity without separation of variables
• P,T,PT, and CPT invariance of Hermitian Hamiltonians
• Pseudo-Hermiticity of an Exactly Solvable Two-Dimensional Model

Cited by in corpus (14)

• Pseudo-Hermitian Representation of Quantum Mechanics
• A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points
• Mathematical and physical aspects of complex symmetric operators
• A Hamiltonian Formulation of the Pais-Uhlenbeck Oscillator that Yields a Stable and Unitary Quantum System
• $J$-self-adjoint operators with $\mathcal{C}$-symmetries: extension theory approach
• Quadratic PT-symmetric operators with real spectrum and similarity to self-adjoint operators
• Non-Hermitian spectral effects in a PT-symmetric waveguide
• Bounds on the spectrum and reducing subspaces of a J-self-adjoint operator
• On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)
• PT Symmetric Schrödinger Operators: Reality of the Perturbed Eigenvalues
• PT-symmetry and Schrödinger operators. The double well case
• Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry
• Three-Dimensional Shape Invariant Non-Separable Model With Equidistant Spectrum
• Non-real Eigenvalues for PT-Symmetric Double Wells