Self-adjoint oscillator operator from a modified factorization

Abstract

"By using an alternative factorization, we obtain a self-adjoint oscillator operator of the form Lδ=ddx(pδ(x)ddx)−(x2pδ(x)+pδ(x)−1), where pδ(x)=1+δe−x2, with δ∈(−1,∞) an arbitrary real factorization parameter. At positive values of δ, this operator interpolates between the quantum harmonic oscillator Hamiltonian for δ=0 and a scaled Hermite operator at high values of δ. For the negative values of δ, the eigenfunctions look like deformed quantum mechanical Hermite functions. Possible applications are mentioned."