Asymptotic spectral analysis in colliding leaky quantum layers

Abstract

We consider the Schrödinger operator with a complex delta interaction supported by two parallel hypersurfaces in the Euclidean space of any dimension. We analyse spectral properties of the system in the limit when the distance between the hypersurfaces tends to zero. We establish the norm-resolvent convergence to a limiting operator and derive first-order corrections for the corresponding eigenvalues.

Spectral asymptotics induced by approaching and diverging planar circles

“…The last component of (3.9) reflects contribution of the curvature to the first correction term. More general situation shows a presence of the first mean curvature in eigenvalue asymptotics, cf [13]. Furthermore, let us note that a contribution of the first mean curvature in spectral asymptotics has been recently showed in related problems, see [14], [15] and [16].…”

“…Let us mention that the above formula describes a very particular case of the class considered in the forthcoming paper [13]. In this paper the spectral asymptotics for approaching hypersurfaces in R d is analyzed.…”

“…In this paper the spectral asymptotics for approaching hypersurfaces in R d is analyzed. The method developed in [13] allows to reconstruct asymptotics of eigenvalues by means of the "unperturbed" eigenfunctions. The technics enables generalization for complex coupling constants.…”

Soft quantum waveguides with an explicit cut-locus

“…This follows directly from the argument derived in the proof of [8, 10, lemma 2.3]. More detailed analysis of eigenvalues for the model of colliding dots was developed in [9]. Now we are in a position to localize the essential spectrum of H.…”