On the Discrete Spectrum of a Family of Differential Operators

We consider a family $\mathbf{A}_\alpha$ of differential operators in $L^2(\mathbb{R}^2)$ depending on a parameter $\alpha\ge0$. The operator $\mathbf{A}_\alpha$ formally corresponds to the quadratic form
$$ \mathbf{a}_\alpha[U]=\int_{\mathbb{R}^2}\biggl(|U_x|^2+\frac{1}{2}(|U_y|^2 +y^2|U|^2)\biggr)\,dx\,dy +\alpha\int_\mathbb{R}y|U(0,y)|^2\,dy. $$
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 $\mathbf{a}_0$.
The spectral properties of $\mathbf{A}_\alpha$ strongly depend on $\alpha$. In particular, $\sigma(\mathbf{A}_0)=[1/2,\infty)$; for $0<\alpha<\sqrt 2$, finitely many eigenvalues $l_n<1/2$ are added to the spectrum; and for $\alpha>\sqrt2$ (where the quadratic form approach does not apply), the spectrum is purely continuous and coincides with $\mathbb{R}$. We study the asymptotic behavior of the number of eigenvalues as $\alpha\nearrow\sqrt 2$ and reduce this problem to the problem on the spectral asymptotics for a certain Jacobi matrix.