A regular analogue of the Smilansky model: Spectral properties. Issue 2 (October 2017) - Barseghyan, Diana; Exner, Pavel
Abstract : We analyze spectral properties of the operator H = ∂ 2 / ∂x 2 — ∂ 2 / ∂y 2 + ω 2 y 2 – λy 2 V ( xy ) in L 2 (ℝ 2 ), where ω ≠ 0 and V ≥ 0 is a compactly supported and sufficiently regular potential. It is known that the spectrum of H depends on the one-dimensional Schrödinger operator L = — d 2 / dx 2 + ω 2 — λV ( x ) and it changes substantially as infσ( L ) switches sign. We prove that in the critical case, infσ( L ) = 0, the spectrum of H is purely essential and covers the interval [0, ∞). In the subcritical case, inf σ ( L ) > 0, the essential spectrum starts from ω and there is a nonvoid discrete spectrum in the interval [0, ω ). We also derive a bound on the corresponding eigenvalue moments.