On norm resolvent convergence of Schrödinger operators with ${\delta }^{\prime }$-like potentials

Abstract

We address the problem on the right definition of the Schroedinger operator with potential ${\delta }^{\prime }$, where $\delta$ is the Dirac delta-function. Namely, we prove the uniform resolvent convergence of a family of Schroedinger operators with regularized short-range potentials ${\epsilon }^{-2}V(x/\epsilon )$ tending to ${\delta }^{\prime }$ in the distributional sense as $\epsilon \to 0$. In 1986, P. Seba claimed that the limit coincides with the direct sum of free Schroedinger operators on the semi-axes with the Dirichlet boundary condition at the origin, which implies that in dimension one there is no non-trivial Hamiltonians with potential ${\delta }^{\prime }$. Our results demonstrate that, although the above statement is true for many V, for the so-called resonant V the limit operator is defined by the non-trivial interface condition at the origin determined by some spectral characteristics of V. In this resonant case, we show that there is a partial transmission of the wave package for the limiting Hamiltonian.

16 pages, 2 figures. The proof of Lemma 2.1 was corrected.The main results of the paper are unchanged. Other minor changes were made

References in corpus (3)

• Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide
• Coupling in the singular limit of thin quantum waveguides
• Boundary conditions for the states with resonant tunnelling across the ${\delta }^{\prime }$-potential

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