For a function V : R → R that is integrable and compactly supported, we prove the norm resolvent convergence, as ε → 0, of a family S ε of one-dimensional Schrödinger operators on the line of the formthen the functions ε −2 V (x/ε) converge in the sense of distributions as ε → 0 to δ ′ (x), and the limit S 0 of S ε might be considered as a 'physically motivated' interpretation of the one-dimensional Schrödinger operator with potential δ ′ . In 1985,Šeba claimed that the limit operator S 0 is the direct sum of the free Schrödinger operators on positive and negative semi-axes subject to the Dirichlet condition at x = 0, which suggested that in dimension 1 there is no non-trivial Hamiltonian with potential δ ′ . In this paper, we show that in fact S 0 essentially depends on V : although the above results are true generically, in the exceptional (or 'resonant') case, the limit S 0 is non-trivial and is determined by the properties of an auxiliary Sturm-Liouville spectral problem associated with V . We then set V (ξ) = αΨ(ξ) with a fixed Ψ and show that there exists a countable set of resonances {α k } ∞ k=−∞ for which a partial transmission of the wave package occurs for S 0 .

For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as ε → 0, of a family S ε of one-dimensional Schrödinger operators on the line of the formUnder certain conditions the functions ε −2 V (x/ε) converge in the sense of distributions as ε → 0 to δ ′ (x), and then the limit S 0 of S ε might be considered as a "physically motivated" interpretation of the one-dimensional Schrödinger operator with potential δ ′ .

Abstract. For a function V : R → R that is integrable and compactly supported, we prove the norm resolvent convergence, as ε → 0, of a family S ε of one-dimensional Schrödinger operators on the line of the formthen the functions ε −2 V (x/ε) converge in the sense of distributions as ε → 0 to δ ′ (x), and the limit S 0 of S ε might be considered as a 'physically motivated' interpretation of the one-dimensional Schrödinger operator with potential δ ′ . In 1985,Šeba claimed that the limit operator S 0 is the direct sum of the free Schrödinger operators on positive and negative semi-axes subject to the Dirichlet condition at x = 0, which suggested that in dimension 1 there is no non-trivial Hamiltonian with potential δ ′ . In this paper, we show that in fact S 0 essentially depends on V : although the above results are true generically, in the exceptional (or 'resonant') case, the limit S 0 is non-trivial and is determined by the properties of an auxiliary Sturm-Liouville spectral problem associated with V . We then set V (ξ) = αΨ(ξ) with a fixed Ψ and show that there exists a countable set of resonances {α k } ∞ k=−∞ for which a partial transmission of the wave package occurs for S 0 .

We consider the vibrations of a membrane that contains a very thin and heavy inclusion around a curve γ. We assume that the membrane occupies a domain Ω of ℝ2. The inclusion occupies a layer-like domain ωε of width 2ε and it has a density of order O(ε-m). The density is of order O(1) outside this inclusion ωε, the concentrated mass around the curve γ. ε and m are positive parameters, ε∈(0,1) and m>2. We set m=3 and show that low, middle and high frequency vibrations are necessary in order to describe the asymptotic behavior of the vibrations of the whole membrane. We study the asymptotic behavior, as ε→0, of these frequencies and of the corresponding eigenfunctions.

The threshold behaviour of negative eigenvalues for Schr\"{o}dinger operators of the type \[ H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda\alpha_\lambda V(\alpha_\lambda \cdot) \] is considered. The potentials $U$ and $V$ are real-valued bounded functions of compact support, $\lambda$ is a positive parameter, and positive sequence $\alpha_\lambda$ has a finite or infinite limit as $\lambda\to 0$. Under certain conditions on the potentials there exists a bound state of $H_\lambda$ which is absorbed at the bottom of the continuous spectrum. For several cases of the limiting behaviour of sequence $\alpha_\lambda$, asymptotic formulas for the bound states are proved and the first order terms are computed explicitly.