On the Discrete Spectrum of a Family of Differential Operators

Abstract

We consider a family A _α of differential operators in L ²(ℝ²) depending on a parameter α≥0. The operator A _α formally corresponds to the quadratic form

$$a_\alpha [U] = \int_{\mathbb{R}^2 } {(|U_x |^2 + \frac{1}{2}(|U_y |^2 + y^2 |U|^2 ))dx{\text{ }}dy + \alpha \int_\mathbb{R} {y|U(0,y)|^2 dy.} }$$

The perturbation determined by the second term in this sum is only relatively bounded but not relatively compact with respect to the unperturbed quadratic form a ₀.

The spectral properties of A _α strongly depend on α. In particular, σ(A ₀)=[1/2,∞); for 0<α<$\sqrt 2$, finitely many eigenvalues λ_n < 1/2 are added to the spectrum; and for α> $\sqrt 2$ (where the quadratic form approach does not apply), the spectrum is purely continuous and coincides with ℝ. We study the asymptotic behavior of the number of eigenvalues as α↗$\sqrt 2$ and reduce this problem to the problem on the spectral asymptotics for a certain Jacobi matrix.