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

For a densely defined self-adjoint operator HH in Hilbert space FF the operator exp⁡(−itH)exp⁡(−itH) is the evolution operator for the Schrödinger equation iψt′=Hψ, i.e. if ψ(0,x)=ψ0(x)ψ(0,x)=ψ0(x) then ψ(t,x)=(exp⁡(−itH)ψ0)(x)ψ(t,x)=(exp⁡(−itH)ψ0)(x) for x∈Qx∈Q. The space FF here is the space of wave functions ψ defined on an abstract space Q  , the configuration space of a quantum system, and HH is the Hamiltonian of the system. In this paper the operator exp⁡(−itH)exp⁡(−itH) for all real values of t   is expressed in terms of the family of self-adjoint bounded operators S(t)S(t), t≥0t≥0, which is Chernoff-tangent to the operator −H−H. One can take S(t)=exp⁡(−tH)S(t)=exp⁡(−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 FF so it can be used in a wider context due to its generality. Two examples of application are provided.