Abstract. We study the eigenvalue spectrum of Dirichlet Laplacians which model quantum waveguides associated with tubular regions outside of a bounded domain. Intuitively, our principal new result in two dimensions asserts that any domain Ω obtained by adding an arbitrarily small "bump" to the tube Ω 0 = R × (0, 1) (i.e., Ω Ω 0 , Ω ⊂ R 2 open and connected, Ω = Ω 0 outside a bounded region) produces at least one positive eigenvalue below the essential spectrum [π 2 , ∞) of the Dirichlet Laplacian −∆ D Ω . For |Ω\Ω 0 | sufficiently small (| . | abbreviating Lebesgue measure), we prove uniqueness of the ground state E Ω of −∆ D Ω and derive the "weak coupling" result E Ω = π 2 − π 4 |Ω\Ω 0 | 2 + O(|Ω\Ω 0 | 3 ) using a Birman-Schwinger-type analysis. As a corollary of these results we obtain the following surprising fact: Starting from the tube Ω 0 with Dirichlet boundary conditions at ∂Ω 0 , replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment (a, b) × {1}, a < b, of ∂Ω 0 . If H(a, b) denotes the resulting Laplace operator in L 2 (Ω 0 ), then H(a, b) has a discrete eigenvalue in [π 2 /4, π 2 ) no matter how small |b − a| > 0 is. §1. Introduction Our goal in this paper is to study the bound state spectra of the Dirichlet Laplacian −∆D Ω for open regions Ω ⊂ R n which are tubes outside of a bounded region (quantum waveguides). (Following the traditional notation in quantum physics, we denote the Laplacian by −∆ as opposed to ∆ in the following.) In particular, let Ω 0 ⊂ R 2 be defined byConsider open connected sets Ω such that:Because of condition (i),Then one of our main goals will be to prove
We consider Schrödinger operators H in L2(Rn), n ∈ N, with countably infinitely many local singularities of the potential which are separated from each other by a positive distance. It is proved that due to locality each singularity yields a separate contribution to the deficiency index of H. In the special case where the singularities are pointlike and the potential exhibits certain symmetries near these points we give an explicit construction of self-adjoint boundary conditions.
Abstract. Combining algebro-geometric methods and factorization techniques for finite difference expressions we provide a complete and self-contained treatment of all real-valued quasi-periodic finite-gap solutions of both the Toda and Kac-van Moerbeke hierarchies.In order to obtain our principal new result, the algebro-geometric finitegap solutions of the Kac-van Moerbeke hierarchy, we employ particular commutation methods in connection with Miura-type transformations which enable us to transfer whole classes of solutions (such as finite-gap solutions) from the Toda hierarchy to its modified counterpart, the Kac-van Moerbeke hierarchy, and vice versa.
The free Dirac operator defined on composite one-dimensional structures consisting of finitely many half-lines and intervals is investigated. The influence of the connection points between the constituents is modeled by transition conditions for the wave functions or equivalently by different self-adjoint extensions of the Dirac operator. General relations between the parameters of the extensions and the eigenvalues resp. the scattering coefficients are derived and then applied to the cases of a bundle of half-lines, a point defect, a branching line, and an eye-shaped structure.
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