Self-adjoint oscillator operator from a modified factorization

Abstract

By using an alternative factorization, we obtain a self-adjoint oscillator operator of the form L δ = d d x ( p δ ( x ) d d x ) − ( x 2 p δ ( x ) + p δ ( x ) − 1 ) , where p δ ( x ) = 1 + δ e − x 2 , 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.