On norm resolvent convergence of Schr\'odinger operators with $\delta'$-like potentials
| Autor: | Golovaty, Yu. D., Hryniv, R. O. |
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| Rok vydání: | 2009 |
| Předmět: | |
| Zdroj: | J. Phys. A: Math. Theor. 43 (2010) 155204 (14pp) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1751-8113/43/15/155204 |
| Popis: | 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. Comment: 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 |
| Databáze: | arXiv |
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