1. The approach mentioned here originates in work by K. Ito. Extensive references are in [1], [2]. For recent work in connection with other definitions see [5],[6] (who also discusses the relation with work by Cameron and Storvick) and [7]. There are other methods of defining Feynman path integrals e.g. A) Time division combined with path approximation (e.g. [5] resp. [22]); B) Reduction to probabilistic methods (e.g. [23]); C) Use of Poisson integrals (see e.g. [24]); D) Various combinations and extensions. Full references will be given in [25].
2. See [3], [4], [8]. The derivation of asymptotic expansions for the solution of Schrödinger equation has been discussed also by methods of p.d.e. by Maslov, Duistermaat, Fujiwara (e.g. [22]) and others. There is a rich literature concerning compact manifolds (see e.g. [8]). The basic ideas for writing in the WKBJ approach the leading term in the trace formula for Schrödinger operators goes back to M. Gutzwiller (for discussions:and references, also in connection with quantum fields see e.g. [9]), The approach to the leading term as h → O is also discussed from a mathematical definition of Feynman path integrals in [10], [11]. The trace formula has been proven recently by methods of p.d.e. by Chazarain [12]. Recent very interesting work on the structure of classical orbits with prescribed period has been given by C. Conley and E. Zehnder [13].
3. A note on probabilitstic methods related to the Feynman path integral methods. We shall mention some recent related work which uses probabilistic path integrals rather than Feynman path integrals. 1. Asymptotics-of-function-space-integrals-with-respect-to-Gaussian measures There is a tradition (Donsker, Schilder, Pinkus, Varadhan) in such studies, see e.g. [14]. Recent results related to the one in 12) have been obtained by R. Ellis, J. Rosen concerning the case where the formal complex Gaussian measure of (2), is replaced by a Gaussian probability measure. Here instead of the stationary points of the phase one has to look at minimum points. The expansion corresponds to the one given in our case. The results have applications to the convergence of associated random variables and by this to some problems in classical statistical mechanics (for a very recent result see [15]). Related methods with other applications have been used recently by I. Davies and A. Truman [16] (see also [11] for manifolds and e.g. [17], for quantum fields). 2. Methods of stochastic-equations. The methods have been used in connection with solutions of the Schrödinger equations [18] and their eigenvalues and eigenfunctions [12], [19]. Recently an interesting relation between heuristic Feynman path integrals for the Coulomb problem on ℝ3 and Feynman path integrals for the harmonic oscillator on ℝ4 has been discovered by I.H. Duru and H. Kleinert [20]. Ph. Blanchard and M. Sirugue [21] have proven the validity of this relation replacing Feynman path integrals by Wiener ones.
4. S. Albeverio, R. Høegh-Krohn, Mathematical Theory of Feynman Path Integrals, Lecture Notes in Maths. 523, Springer (1976).
5. S. Albeverio, R. Høegh-Krohn, pp. 3-57 in “Feynman Path Integrals”, Ed. S. Albeverio et al., Lect. Notes in Phys. 106, Springer (1979).