A Method to Calculate a Delta Function of the Hamiltonian by the Suzuki-Trotter Decomposition

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We propose a new method to calculate expectation values of a delta function of the Hamiltonian, < Ψ δ(H - E) Ψ>. Since the delta function can be replaced with a Gaussian function, we evaluate < Ψ √βπ e-β(H - E)² Ψ> with large βadopting the Suzuki-Trotter decomposition. Errors of the approximate calculations with the finite Trotter number Nₜ are estimated to be O(1/Nₜᵏ) for the Kth-order decomposition. The distinct advantage of this method is that the convergence is guaranteed even when the state Ψ> 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.