  We consider the Schr{\"o}dinger operator $-\Delta +V(x)$ in $L^2({\bf R}^3)$
with a real short-range (integrable) potential $V$. Using the associated
Fredholm determinant, we present new trace formulas, in particular, the ones in
terms of resonances and eigenvalues only. We also derive expressions of the
Dirichlet integral, and the scattering phase. The proof is based on the change
of view points for the above mentioned problems from the operator theory to the
complex analytic (entire) function theory.
