A regular analogue of the Smilansky model: Spectral properties

We analyze spectral properties of the operator H = ∂2/∂x2 - ∂2/∂y2 + ω2y2 - λy2V(xy) in L2(ℝ2), 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 = -d2/dx2 + ω2 - 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.