Soft quantum waveguides in three dimensions

We discuss a three-dimensional soft quantum waveguide, in other words, Schr\"odinger operator $\R^3$ with an attractive potential supported by infinite tube and keeping its transverse profile fixed. show that if the is asymptotically straight, distance between ends unbounded, twist satisfies so-called Tang condition, esential spectrum not affected smooth bends. Furthermore, we derive sufficient expressed terms of geometry, for discrete such to be nonempty.