We consider a family Aα of differential operators in L2(ℝ2) depending on a parameter α ≥ 0. The operator Aα formally corresponds to the quadratic form
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 a0.
The spectral properties of Aα strongly depend on α. In particular, σ(A0) = [1/2, ∞); for 0 < α < √2, finitely many eigenvalues λn < 1/2 are added to the spectrum; and for α > √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 ↗ √2 and reduce this problem to the problem on the spectral asymptotics for a certain Jacobi matrix.