Mathematics – Spectral Theory
Scientific paper
2009-11-05
J. Phys. A: Math. Theor. 43 (2010) 155204 (14pp)
Mathematics
Spectral Theory
16 pages, 2 figures. The proof of Lemma 2.1 was corrected.The main results of the paper are unchanged. Other minor changes wer
Scientific paper
10.1088/1751-8113/43/15/155204
We address the problem on the right definition of the Schroedinger operator with potential $\delta'$, 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 $\delta'$ 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 $\delta'$. 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.
Golovaty Yu. D.
Hryniv Rostyslav O.
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