Gap asymptotics in a weakly bent leaky quantum wire

Abstract

Abstract. The main question studied in this paper concerns the weak-coupling behavior of the geometrically induced bound states of singular Schrödinger operators with an attractive δ interaction supported by a planar, asymptotically straight curve Γ. We demonstrate that if Γ is only slightly bent or weakly deformed, then there is a single eigenvalue and the gap between it and the continuum threshold is in the leading order proportional to the fourth power of the bending angle, or the deformation parameter. For comparison, we analyze the behavior of a general geometrical induced eigenvalue in the situation when one of the curve asymptotes is wiggled.

Variational proof of the existence of eigenvalues for star graphs

“…The two-dimensional Hamiltonians with interactions supported by curves have become a prominent class of solvable models of quantum mechanics [8] and are usually referred to as leaky quantum graphs. A summary of various questions and results in the spectral theory of such operators can be found in the review by Exner [7], and for the most recent developments we refer to the papers [1,5,10,14,15,17,18] and to Chapter 10 in the recent monograph by Exner and Kovařík [11].…”

Spectral properties of soft quantum waveguides

“…One such question is whether the gap between the ground state and the continuum in slightly bent potential channels would be proportional to the fourth power of the bending angle as it is the case for hard-wall tubes [EK15, Thm. 6.3] and leaky wires [EKo15]. (vi) Various spectral optimization problems come to mind.…”

Bound states for Laplacian perturbed by varying potential supportedby line in R3

“…The Krein-like formula was fully described in [3]. There are a number of works, which use this procedure for different types of curves [8,[10][11][12]19,20]. Existing results also include works considering the case of R 2 and codimension one [1,4,5] on bent curves or loops [7,9], as well as finite curves and different types of interaction along the curve [13][14][15].…”