Quantum bound states in open geometries

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Abstract
We study the existence of bound states of two-dimensional Helmholtz equations with Dirichlet boundary conditions in open domains. The particular geometry studied here corresponds to a ‘‘broken strip,’’ namely a bent pipe open at both ends and made of two channels of equal width intersecting at an arbitrary nonzero angle. We prove the existence of at least one bound state for any arbitrarily small angle, and more than one bound state for sharp broken strips. For small angles, the binding energy is found to be quartic with respect to the bending angle. We calculate this asymptotic binding analytically. Other geometries (like that of crossing wires) can be handled with our technique.