We discuss a modification of Smilansky model in which a singular potential
`channel' is replaced by a regular, below unbounded potential which shrinks as
it becomes deeper. We demonstrate that, similarly to the original model, such a
system exhibits a spectral transition with respect to the coupling constant,
and determine the critical value above which a new spectral branch opens. The
result is generalized to situations with multiple potential `channels'.Comment: LaTeX, 17 page
We analyze two-dimensional Schrödinger operators with the potential |xy| p − λ(x 2 + y 2 ) p/(p+2) where p ≥ 1 and λ ≥ 0. We show that there is a critical value of λ such that the spectrum for λ < λ crit is below bounded and purely discrete, while for λ > λ crit it is unbounded from below. In the subcritical case we prove upper and lower bounds for the eigenvalue sums.
We consider a family of quantum graphs {(Γ, A ε )} ε>0 , where Γ is a Z n -periodic metric graph and the periodic Hamiltonian A ε is defined by the operation −ε −1 d 2 dx 2 on the edges of Γ and either δ -type conditions or the Kirchhoff conditions at its vertices. Here ε > 0 is a small parameter. We show that the spectrum of A ε has at least m gaps as ε → 0 (m ∈ N is a predefined number), moreover the location of these gaps can be nicely controlled via a suitable choice of the geometry of Γ and of coupling constants involved in δ -type conditions.
Abstract. We analyze two-dimensional Schrödinger operators with the potential |xy| p − λ(x 2 + y 2 ) p/(p+2) where p ≥ 1 and λ ≥ 0, which exhibit an abrupt change of its spectral properties at a critical value of the coupling constant λ. We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for λ below the critical value the spectrum is purely discrete and we establish a LiebThirring-type bound on its moments. In the critical case the essential spectrum covers the positive halfline while the negative spectrum can be only discrete, we demonstrate numerically the existence of a ground state eigenvalue.
Abstract. We analyze spectral properties of two mutually related families of magnetic Schrödinger operators,, with the parameters ω > 0 and λ < 0, where A is a vector potential corresponding to a homogeneous magnetic field perpendicular to the plane and V is a regular nonnegative and compactly supported potential. We show that the spectral properties of the operators depend crucially on the one-dimensional, respectively. Depending on whether the operators L and L(V ) are positive or not, the spectrum of H Sm (A) and H(V ) exhibits a sharp transition.
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