Remarks on convergence of the Feynman path integrals

Feynman path integrals as analysis on path space by time slicing approximation

“…We can also prove that the fundamental theorem of calculus holds in Feynman path integral, i.e., There are some mathematical works which prove the time slicing approximation of (1.1) converges uniformly with respect to (x, x 0 ) on compact subsets of R 2d . See D. Fujiwara [4,[6][7][8], H. Kitada and H. Kumano-go [14], K. Yajima [22], N. Kumano-go [17], D. Fujiwara and T. Tsuchida [10], and W. Ichinose [11]. However all these works treated (1.1), that is the particular case of (1.4) with F [γ ] ≡ 1.…”

Interaction of Coherent States for Hartree Equations

“…These are two linear equations with time dependent potentials, which are polynomial in y, of degree (at most) two. The following result is classical, see for example [17,18,36]: a 2 ). In addition, the following conservations hold:…”

“…First, we recall that as a consequence of [17,18], the system for the envelopes in the linear case is well-posed in k : k ∈ N, and a ∈ k . Then (2.3) has a unique solution u ∈ C(R + ; k ).…”

On smoothing property of Schrödinger propagators

“…For operators of the form (2.12) a general theory was developed by Asada and Fujiwara [1] and Fujiwara [5] and the proofs of (1) to (3) of the following lemma can be found there. We denote by Symo the set of symbols p(t, s, x, ~, y) with the parameters t and s such that ~ ~ ~…”