Abstract
Abstract. The present talk deals with properties of a quantum particle interacting with a potential having the form of a potential ‘ditch’, or an array of potential wells; we discuss how can be the geometry of the potential support reflected in the spectrum of the corresponding Hamiltonian. In particular, we provide sufficient conditions under which a bend of such a potential guide or array gives rise to a non-empty discrete spectrum; we employ different techniques: the Birman-Schwinger method [1,3] or a direct variational argument based on construction of a suitable trial function [2]. We also address the problem of ground state optimalization in case of loop-shaped configurations [3,4].
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