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

Original · EN

For a densely defined self-adjoint operator H in Hilbert space F the operator (-itH) is the evolution operator for the Schrödinger equation iψ'ₜ=Hψ, i.e. if ψ(0,x)=ψ₀(x) then ψ(t,x)=((-itH)ψ₀)(x) for x∈ Q. The space F here is the space of wave functions ψ defined on an abstract space Q, the configuration space of a quantum system, and H is the Hamiltonian of the system. In this paper the operator (-itH) for all real values of t is expressed in terms of the family of self-adjoint bounded operators S(t), t≥ 0, which is Chernoff-tangent to the operator -H. One can take S(t)=(-tH), 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 F so it can be used in a wider context due to its generality. Two examples of application are provided.