The behavior of the discrete spectrum of the Schr"odinger operator , in quite a general setting, up to a large extent is determined by the behavior of the corresponding heat kernel as and . If this behavior is powerlike, i.e., [|P(t;\cdot,\cdot)|{L^\infty}=O(t^{-\delta/2}),\ t\to 0;\qquad |P(t;\cdot,\cdot)|{L^\infty}=O(t^{-D/2}),\ t\to\infty,] then it is natural to call the exponents "{\it the local dimension}" and "{\it the dimension at infinity}" respectively. The character of spectral estimates depends on the relation between these dimensions. In the paper we analyze the case where $\delta
Diego Ghezzi, Charles-Henri Puncho Jérôme Vila