The even-order harmonic oscillator perturbed by a decreasing scalar potential

Abstract

This paper continues a previous work studying the perturbation L = H + V on ℝ, with H = (-1)h ½d2h/dx2h + x2h, where h ∈ ℕ*, and V is a decreasing scalar potential. It is assumed that |V(n)(x)| ≤ cn(1 + x2)-s/2 for x ∈ ℝ, s ∈ ℝ*+ \ {1}, and n ∈ ℕ. Let λk denote the kth eigenvalue of H, and suppose the eigenvalues of L near λk can be expressed as λk + μk. The authors prove an asymptotic formula for the fluctuation {μk}, which is given by a transformation of V. This paper specifically examines the case where s = 1.