Abstract
We discuss a three-dimensional soft quantum waveguide, in other words, Schrödinger operator in R3 with an attractive potential supported by an infinite tube and by keeping its transverse profile fixed. We show that if the tube is asymptotically straight, the distance between its ends is unbounded, and its twist satisfies the so-called Tang condition, the essential spectrum is not affected by smooth bends. Furthermore, we derive a sufficient condition, expressed in terms of the tube geometry, for the discrete spectrum of such an operator to be nonempty.
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