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

Abstract

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.

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

“…Definition 1.2. (First introduced in [11]) Let us say that G is Chernofftangent to L iff the following conditions of Chernoff tangency (CT) hold:…”

“…For the case of Schrödinger equation (L = iH, where H is a self-adjoint operator equal to Hamiltonian with inverse sign, H = −H) another approach was proposed in 2014 [53] (published with full proof in 2016 [11]). Proposed idea is as follows: we find S that is Chernoff-tangent to H (e.g.…”

“…Formal statement follows. ; new wording of theorem 3.1 from [11]). Let F be a complex Hilbert space and let Dom(H) ⊂ F be its dense linear subspace.…”

“…In short, theorem states the following: if S is Chernoff tangent to H, and operators H and S(t) are self-adjoint, then R(t) = e i(S(t)−I) is Chernoff-equivalent to (e itH ) t≥0 . The difference between S and W is that S(0) = I and W (0) = 0 which makes expression (9) simpler than the original form in theorem 3.1 in [11]. Note that we have NOT used the norm bound condition (N) for S here, but still achieved (N) for R, so this approach is more flexible than the standard procedure of finding a family of integral operators that is Chernoff-equivalent to C 0 -semigroup e itH t≥0 .…”

“…M.S. Buzinov in 2015 has obtained [59,58,11] Feynman formulas for heattype evolution equation with natural power of Laplacian on the right-hand side of the equation, but for corresponding Schrödinger equation he only constructed quasi-Feynman formulas using theorem 1.3. See also section 6 of [46] where authors provide solution for a particular case of Schrödinger equation with constant coefficients and derivative of 6-th order in the Hamiltonian.…”