Symmetric non-self-adjoint operators, diagonalizable and non-diagonalizable, with a real discrete spectrum

Abstract

Consider in , the operator family . H0 = a*1a1 + ⋅ ⋅ ⋅ + a*dad + d/2 is the quantum harmonic oscillator with rational frequencies, W is a symmetric bounded potential, and g is a real coupling constant. We show that if |g| < ρ, ρ being an explicitly determined constant, the spectrum of H(g) is real and discrete. Moreover we show that the operator has a real discrete spectrum but is not diagonalizable.