Yu. Golovaty

We consider one dimensional Schrödinger operators \begin{equation*} H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda V_\lambda \end{equation*} with nonlinear dependence on the parameter $\lambda$ and study the small $\lambda$ behavior of eigenvalues. Potentials $U$ and $V_\lambda$ are real-valued bounded functions of compact support. Under some assumptions on $U$ and $V_\lambda$, we prove the existence of a negative eigenvalue that is absorbed at the bottom of the continuous spectrum as $\lambda\to 0$. We also construct two-term asymptotic formulas for the threshold eigenvalues.

For real functions $\Phi$ and $\Psi$ that are integrable and compactly supported, we prove the norm resolvent convergence, as $\varepsilon\to0$, of a family $S_\varepsilon$ of one-dimensional Schrödinger operators on the line of the form $$ S_\varepsilon= -\frac{d^2}{d x^2}+\alpha\varepsilon^{-2}\Phi(\varepsilon^{-1}x)+\beta\varepsilon^{-1}\Psi(\varepsilon^{-1}x). $$ The limit results are shape-dependent: without reference to the convergence of potentials in the sense of distributions the limit operator $S_0$ exists and strongly depends on the pair $(\Phi,\Psi)$. A class of nontrivial point interactions which are formally related the pseudo-Hamiltonian $-\frac{d^2}{dx^2}+\alpha\delta'(x)+\beta\delta(x)$ is singled out. The limit behavior, as $\varepsilon\to 0$, of the scattering data for such potentials is also described.