1.

E. Zeidler, Quantum Field theory I Basics in Mathematics and Physics, Springer 2006.

http://www.mis.mpg.de/zeidler/qft.html
is a book I highly recommend. It is the first volume of a sequence, of which not all volumes have been published yet. This volume gives an overview over the main mathematical techniques used in quantum physics, in a way that you cannot find anywhere else.

It is a mix of rigorous mathematics and intuitive explanation, and tries to build ''A bridge between mathematiciands and physicists'' as the subtitle says. It makes very interesting reading if you know already enough math and physics, and gives plenty of references as entry points to the literature for topics on which your background is meager.

As regards to your request for high level mathematics (in the specific form of pseudo-differential operators, etc.), Zeidler discusses - as Section 12.5 - on 28 (of 958 total) pages microlocal analysis and its use, though there is only two pages specifically devoted to PDO (p.728-729), but he says there (and emphasizes) that ''Fourier integral operators play a fundamental role in quantum field theory for describing the propagation of physical effects'' - so you can expect that they play a more prominent role in the volumes to come.

But, of course, PDO are implicit in all serious high level mathematical work on quantum mechanics even without mentioning them explicitly, as for example the Hamiltonian in the interaction representation, $H_{int}=e^{-itH_0}He^{itH_0}$, is a PDO. Work on Wigner transforms is work on PDOs, etc..

2.

Other books using PDO, much more specialized:

G. B. Folland, Harmonic analysis in phase space

A.L. Carey, Motives, quantum field theory, and pseudodifferential operators

A. Juengel, Transport equations for semiconductors

C. Cercignani and E. Gabetta, Transport phenomena and kinetic theory

N.P. Landsman, Mathematical topics between classical and quantum mechanics

M. de Gosson, Symplectic geometry and quantum mechanics

P. Zhang, Wigner measure and semiclassical limits of nonlinear Schroedinger equations

3.

Finally, as an example of a book that ''is strictly supported by mathematics (given a set of mathematically described axioms, the author develops the theory using mathematics as a main tool)'', I can offer my own book

A. Neumaier and D. Westra,

Classical and Quantum Mechanics via Lie algebras.

This post imported from StackExchange Physics at 2014-03-24 04:50 (UCT), posted by SE-user Arnold Neumaier