Quasi-Feynman formulas -- a method of obtaining the evolution operator for the Schroedinger equation

Abstract

For a densely defined self-adjoint operator $\mathcal{H}$ in Hilbert space $\mathcal{F}$ the operator $\exp (-it\mathcal{H})$ is the evolution operator for the Schrödinger equation $i{\psi }_t^{\prime }=\mathcal{H}\psi$, i.e. if $\psi (0,x)={\psi }_0(x)$ then $\psi (t,x)=(\exp (-it\mathcal{H}){\psi }_0)(x)$ for $x\in Q.$ The space $\mathcal{F}$ here is the space of wave functions $\psi$ defined on an abstract space $Q$, the configuration space of a quantum system, and $\mathcal{H}$ is the Hamiltonian of the system. In this paper the operator $\exp (-it\mathcal{H})$ for all real values of $t$ is expressed in terms of the family of self-adjoint bounded operators $S(t),t\ge 0$, which is Chernoff-tangent to the operator $-\mathcal{H}$. One can take $S(t)=\exp (-t\mathcal{H})$, or use other, simple families $S$ that are listed in the paper. The main theorem is proven on the level of semigroups of bounded operators in $\mathcal{F}$ so it can be used in a wider context due to its generality. Two examples of application are provided.

24 pages

References in corpus (2)

• Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations
• Solution to a parabolic differential equation in Hilbert space via Feynman formula - parts I and II

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• Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker-Planck-Kolmogorov equations
• Formulas that represent Cauchy problem solution for momentum and position Schrödinger equation
• Speed of convergence of Chernoff approximations to solutions of evolution equations
• Quasi-Feynman formulas for a Schroedinger equation with a Hamiltonian equal to a finite sum of operators