Asymptotics of spectrum of a harmonic oscillator perturbed by a nonsmooth potential

Abstract

1. In this paper we study the operator H = H0 + V in L2(R), where H0 = −d2/dx2 + x2, V is the operator of multiplication by a real, measurable, decreasing for x → ∞ function. It is well-known (see, e.g., [1], p. 326) that the spectrum of the operator H0 consists of the numbers 2n+ 1, and the corresponding normalized eigenfunctions are φn(x) = Hn(x)e−x 2/2/ √ 2nn! √ π, n = 0, 1, 2, . . . , where Hn(x) are the Chebyshev–Hermite polynomials. The asymptotics of eigenvalues of the perturbed operator H = H0 + V for smooth decreasing at infinity functions was first studied in detail in [2], where an etalon solution was obtained with the help of the Airy functions ([3], p. 377). Since V (x) is not necessarily smooth, one cannot immediately apply the method of etalon solutions to the function q(x) = x2 + V (x). In this paper we use the apparatus of perturbation theory based on the study of the asymptotic representation of the kernel of the resolvent of the unperturbed operator. Let λn stand for eigenvalues of the operator H0, let Pn denote the corresponding projectors onto eigensubspaces, and let R0(λ) be the resolvent of the operator H0, R0(λ) = (H0 − λ)−1. According to [4], if V satisfies the condition