Spectra of three-dimensional cruciform and lattice quantum waveguides

Examples of Plentiful Discrete Spectra in Infinite Spatial Cruciform Quantum Waveguides

“…Quite the same conclusions but by means of a substantively modified approach were made in [3,4] for a rectangular quantum lattice composed of thin circular cylinders as well as for the cruciform waveguide Fig. 2,a, which also has only one point in the discrete spectrum of the Dirichlet problem.…”

Eigenvalue inequalities and absence of threshold resonances for waveguide junctions

“…In [2,3], the intersection of two circular cylinders was considered, and the analysis was more involved. In particular, it was shown using an asymptotic estimate that the conditions of Corollary 4 are satisfied if one chooses a sufficiently big center.…”

“…Recently, some specific star waveguide configurations in two and three dimensions were studied in [2,21,22], and the absence of threshold resonances was shown. One should remark that, in all the cases considered, the condition (2) is not satisfied, and a non-empty discrete spectrum is present.…”

“…One should remark that, in all the cases considered, the condition (2) is not satisfied, and a non-empty discrete spectrum is present. The aim of the present paper is to state explicitly the main condition used in the constructions of [2,21,22] and then to show how it can be applied to the analysis of more general geometric configurations. Our main contribution is as follows:…”

“…As noted above, Proposition 2 is a special case of Theorem 3 with N (Λ) = 0. For further references, let us state explicitly another obvious but important corollary corresponding to N (Λ) = 1, which is essentially the condition used in [2,21,22]: Corollary 4. If the discrete spectrum −∆ Λ D is non-empty and for some center C one has λ 2 (−∆ C DN ) > ν, then −∆ Λ D has a single discrete eigenvalue and no threshold resonance.…”