For a densely defined self-adjoint operator H in Hilbert space F the operator exp(−itH) is the evolution operator for the Schrödinger equation iψThe 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 exp(−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) = 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 F so it can be used in a wider context due to its generality. Two examples of application are provided.
We present a general method of solving the Cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two, one and zero. The method is based on the Chernoff approximation procedure applied to a specially constructed shift operator. It is proven that approximations converge uniformly to the exact solution.
Keywords:Cauchy problem, linear parabolic PDE, approximate solution, shift operator, Chernoff theorem, numerical method 2000 MSC: 35A35, 35C99, 35K15, 35K30
Problem setting and approach proposedConsider x ∈ R 1 , t ≥ 0 and set the Cauchy problem for a second-order parabolic partial differential equationThe coefficients a, b, c, u 0 above are bounded, uniformly continuous functions R 1 → R 1 . This paper is dedicated to deriving of an explicit formula
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.