Discrete Spectrum in the Gaps of the Continuous one in the Large-Coupling-Constant Limit

A Note on the Asymptotic and Threshold Behaviour of Discrete Eigenvalues inside the Spectral Gaps of the Difference Operator with a Periodic Potential

“…We choose our 's such that the gaps are empty when = 0. For > 0, the spectrum of is made up of the continuous part which coincides with that of ( 0 ) together with at most a finite number of eigenvalues inside the gap of ( 0 ) 2 Advances in Mathematical Physics [3,[10][11][12]. Of interest to this work are the conditions on guaranteeing the existence of such eigenvalues and their asymptotic behaviour as varies.…”

Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit

“…in the twodimensional case ( [9], [10]). In recent years an analogous asymptotic analysis of the point spectrum arising in spectral gaps of Schrödinger operators under perturbations has attracted considerable attention; starting from [1], Birman has developed a general framework to study this problem [2], [3], [4], [5], [6], [7]; see also [16].…”

“…in the two-dimensional case ( [9], [10]). In recent years an analogous asymptotic analysis of the point spectrum arising in spectral gaps of Schrödinger operators under perturbations has attracted considerable attention; starting from [1], Birman has developed a general framework to study this problem [2], [3], [4], [5], [6], [7]; see also [16]. More specifically, Sobolev [26] has studied the perturbed periodic one-dimensional Schrödinger operator, showing that for a wide range of power-decaying perturbations, the number of eigenvalues arising in a closed subinterval of a spectral gap of the unperturbed problem is asymptotically given by a quasi-semiclassical formula in which the quasimomentum of the periodic background problem takes the role of the ordinary momentum in the Weyl formula.…”