On norm resolvent convergence of Schrödinger operators with δ′-like potentials

Yu D Golovaty, R O Hryniv
2010 Journal of Physics A: Mathematical and Theoretical  
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
more » ... 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.
doi:10.1088/1751-8113/43/15/155204 fatcat:gdu4i2axurcpvhwde46fiywawy