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We show that there is a family Schrdinger operators with scaled potentials which approximates the '-interaction Hamiltonian in the norm-resolvent sense. This approximation, based on a formal scheme proposed by Cheon and Shigehara, has nontrivial convergence properties which are in several respects opposite to those of the Klauder phenomenon.

The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schrdinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.

No abstract uploaded!

Using the complex coloring method, we present the graphs of the quantum dilogarithm function Gb(z) and visualize its analytic and asymptotic behaviours. In particular we demonstrate the limiting process when the modified Gb(z)→Γ(z) as b→0. We also survey the relations of Gb(z) with different variants of the quantum dilogarithm function.