کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4589610 | 1630606 | 2016 | 18 صفحه PDF | دانلود رایگان |

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.
Journal: Journal of Functional Analysis - Volume 270, Issue 12, 15 June 2016, Pages 4540–4557