Affiliations: Bashkir State Pedagogical University, October St. 3a, 450000 Ufa, Russian Federation. E-mail: borisovdi@yandex.ru | Department of Theoretical Physics, Nuclear Physics Institute ASCR, 25068 Řež, Czech Republic. E-mail: krejcirik@ujf.cas.cz | IKERBASQUE, Basque Foundation for Science, Alameda Urquijo, 36, 5, 48011 Bilbao, Spain
Abstract: The Laplacian in an unbounded tubular neighbourhood of a hyperplane with non-Hermitian complex-symmetric Robin-type boundary conditions is investigated in the limit when the width of the neighbourhood diminishes. We show that the Laplacian converges in a norm resolvent sense to a self-adjoint Schrödinger operator in the hyperplane whose potential is expressed solely in terms of the boundary coupling function. As a consequence, we are able to explain some peculiar spectral properties of the non-Hermitian Laplacian by known results for Schrödinger operators.
Keywords: Laplacian in shrinking tubular neighbourhoods, non-Hermitian Robin boundary conditions, effective Hamiltonian, PT-symmetry, J-self-adjointness, norm-resolvent convergence
