Examples of Plentiful Discrete Spectra in Infinite Spatial Cruciform Quantum Waveguides

Abstract

Spatial cruciform quantum waveguides (the Dirichlet problem for Laplace operator) are constructed such that the total multiplicity of the discrete spectrum exceeds any preassigned number.

Existence of the discrete spectrum in the Fichera layers and crosses of arbitrary dimension

“…This fact, in turn, allows us to manage the same proof for n = 4 etc. The only difference is the following: the set Γ 2 12 is just a segment, while for n > 3 the set Γ n−1 12 is unbounded. Nevertheless the integral…”

“…At the same time, including an additional small geometric parameter into the problem contributes to the construction of an arbitrarily large number of eigenvalues, either by enlarging the junction zone of the waveguides (see, e.g. [10,29]), or by the localization effect near angular points (see, e.g., [2]).…”

“…For some classes of convex unbounded domains this problem is well studied, see, e.g., [3,4,22,24]. For Θ 2 1 and Θ 2 2 it was considered in [23].…”

Rectangular lattices of cylindrical quantum waveguides. I. Spectral problems on a finite cross

“…for which the possible generalizations become substantial. For instance, in accordance with [19,20], in an infinite cross (2) formed by the cylinders ( 12)…”

Criteria for the Absence and Existence of Bounded Solutions at the Threshold Frequency in a Junction of Quantum Waveguides

“…If a cranked waveguide belongs to Ω and is composed of two skewed semiinfinite cylinders which have the cross-sections congruent to ω 1 and meet each other under the angle α ∈ (0, π), then κ ≥ 1 according to a result in [1] and the max-min principle [4,Th 10.2.2]. Furthermore, the papers [1,18,5,6] and [3] give examples of arbitrary large κ in dimension 2 and 3, respectively. We refer the book [8] for a completed review of results on the discrete spectrum of quantum waveguides and their junctions.…”