I am reading Gutzwiller's papers on the relation between Hamiltonian flows and solution to Schrodinger equations.


In the two papers, he gave a semi-classical approximation of the Green's function to the Laplacian


Gutzwiller, M. C.: "The Phase Integral Approximation in Momentum Space and the Bound States of an Atom," J. Math. Phys. 8, 1979-2000 (1967).


Gutzwiller, M. C.: "The Phase Integral Approximation in Momentum Space and the Bound States of an Atom II," J. Math. Phys. 10, 1004-1020 (1969).


and on a third paper, he calculated the trace of the operator


Gutzwiller, M. C.: "Periodic Orbits and Classical Quantization Conditions," J. Math. Phys. 12, 343-358 (1971).


There are quite some literature in mathematics giving related results to the trace formula, but I am wondering is there any literature in mathematics related to the approximation of Green's function ?



semi-classical Green's function
The difference between the semiclassical approximations of the full Green's function and the trace is whether or not you restrict the sum over paths to closed orbits; for a treatment of the semiclassical Green's function (the socalled Van Vleck-Gutzwiller propagator), see for example chapter 10 of David Tannor's [Introduction to Quantum Mechanics](http://depts.washington.edu/chemcrs/bulkdisk/chem551A_win08/notes_Tannor_Chapter_10.pdf) or Maslov and Fedoriuk's [Semiclassical Approximations in Quantum Mechanics](http://books.google.nl/books?id=CEwulgtYT6EC).


