  The main aim of this paper is twofold: (1) revealing a relation between the
counting function N(lambda) (the number of the eigenstates with eigenvalue
smaller than a given number) and the heat kernel K(t), which is still an open
problem in mathematics, and (2) introducing an approach for the calculation of
N(lambda), for there is no effective method for calculating N(lambda) beyond
leading order. We suggest a new expression of N(lambda) which is more suitable
for practical calculations. A renormalization procedure is constructed for
removing the divergences which appear when obtaining N(lambda) from a
nonuniformly convergent expansion of K(t). We calculate N(lambda) for
D-dimensional boxes, three-dimensional balls, and two-dimensional
multiply-connected irregular regions. By the Gauss-Bonnet theorem, we
generalize the simply-connected heat kernel to the multiply-connected case;
this result proves Kac's conjecture on the two-dimensional multiply-connected
heat kernel. The approaches for calculating eigenvalue spectra and state
densities from N(lambda) are introduced.
