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We consider a spinless particle coupled to a quantized Bose field and show that such a system has a ground state for two classes of short-range potentials which are alone too weak to have a zero-energy resonance.

The product formula for perturbations of propagators by Faris is generalized: we show that one can use arbitrary partitions of a given interval and perform the limit uniformly with respect to the partition norm. The results are applied to express solutions of the Schrdinger equation, in particular for some complex and time-dependent potentials, by means of Nelson-type path integrals.

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We study approximations of billiard systems by lattice graphs. It is demonstrated that under natural assumptions the graph wavefunctions approximate solutions of the Schroedinger equation with energy rescaled by the billiard dimension. As an example, we analyze a Sinai billiard with attached leads. The results illustrate emergence of global structures in large quantum graphs and offer interesting comparisons with patterns observed in complex networks of a different nature.