The discrete spectrum of cross-shaped waveguides

Abstract

The discrete spectrum of the Dirichlet problem for the Laplace operator on the union of two circular unit cylinders whose axes intersect at the right angle consists of a single eigenvalue. For the threshold value of the spectral parameter, this problem has no bounded solutions. When the angle between the axes reduces, the multiplicity of the discrete spectrum grows unboundedly.

Eigenvalue inequalities and absence of threshold resonances for waveguide junctions

“…As in the preceding examples, the existence of discrete eigenvalues follows by comparing with broken waveguides. Remark that the operator −∆ C DN is simply the Neumann Laplacian on the unit square, and its second eigenvalue is π 2 = ν, and −∆ Λ D cannot have more than one discrete eigenvalue due to (3). On the other hand, as the strict inequality λ 2 (−∆ C DN ) > ν is not satisfied, the absence of threshold resonances does not follow directly from Corollary 4.…”

“…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.…”

Effective operators for Robin eigenvalues in domains with corners

“…Remark that |A + r B r | = r tan θ, and the operator Q r admits then a separation of variables and is unitarily equivalent to L D ⊗ 1 + 1 ⊗ D r , where D r is the Laplacian on (0, r) with the Dirichlet boundary condition at 0 and the Neumann boundary condition at r, and L D is the one-dimensional Laplacian on the interval (0, r tan θ) with 1-Robin condition at 0 and Dirichlet condition on the other end. Therefore, 2 for large r with any fixed C D ∈ (0, π 2 /4). Therefore, the sought estimate (3.9) becomes equivalent to the existence of C N > 0 for which there holds…”

“…To study the difference B [Ju, Ju] − B [u, u] recall that B [Ju, Ju] = j∈J * (J j u) 2 . Using the elementary inequality (x + y) 2 (1 + ε)x 2 + 2y 2 /ε valid for all x, y ∈ R and ε ∈ (0, 1) we estimate (J j u)…”

“…More precise estimates are available for smooth domains. The lower bound by Lou and Zhu [45] and the upper bound due to Daners and Kennedy [15] imply that if Ω is a bounded C 1 domain, then for each fixed n ∈ N one has E n (R Ω α ) ∼ −α 2 . It seems that a more precise asymptotics was first obtained by Pankrashkin in [52]: it was shown that if Ω ⊂ R 2 is bounded with a C 3 boundary, then E 1 (R Ω α ) = −α 2 − H * α + O(α 2 3 ), where H * is the maximum of the curvature of the boundary.…”