$PT$ 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 (2)

• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties
• Pseudo-Hermiticity of an Exactly Solvable Two-Dimensional Model

Cited by in corpus (6)

• A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points
• Mathematical and physical aspects of complex symmetric operators
• Non-Hermitian spectral effects in a PT-symmetric waveguide
• PT Symmetric Schrödinger Operators: Reality of the Perturbed Eigenvalues
• Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry
• PT-symmetry and Schrödinger operators. The double well case