  We propose a new method to calculate expectation values of a delta function
of the Hamiltonian, < \Psi \mid \delta(\hat{H} - E)\mid \Psi >. Since the delta
function can be replaced with a Gaussian function, we evaluate < \Psi \mid
\sqrt{\frac{\beta}{\pi}} e^{-\beta (\hat{H} - E)^2} \mid \Psi > with large
\beta adopting the Suzuki-Trotter decomposition. Errors of the approximate
calculations with the finite Trotter number N_t are estimated to be O(1/N_t^K)
for the Kth-order decomposition. The distinct advantage of this method is that
the convergence is guaranteed even when the state \mid \Psi > contains the
eigenstates whose energies spread over the wide range. In this paper we give a
full description of our method within the quantum mechanical physics and
present the numerical results for the harmonic oscillator problems in one- and
three-dimensional space.
