A book on quantum mechanics supported by the high-level mathematics I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators etc, certainly the modern mathematics). If there isn't something similar please give me a reference to the book that is strictly supported by mathematics (given a set of mathematically descripted axioms author develops the theory using mathematics as a main tool).
 A: A commonly cited classic that might be appropriate for you is Reed & Simon, the set. Be prepared for sticker shock. I'm not sure if that is modern enough for you, however.
The four volumes develop all the functional analysis needed for quantum mechanics and quantum field theory, but also cover a lot of the ground typical mathematical physics texts (such as the 4 volumes of Thirring) cover - and it is definitely more rigorous than Thirring, and easier to read.
A: There are also two books by the St.-Peterburg school which could be worth looking at:

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*L.A. Takhtajan, Quantum Mechanics for Mathematicians
and an older one

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*L.D. Faddeev, O.A. Yakubovskii, Lectures on Quantum Mechanics for Mathematics Students
Takhtajan's book is more advanced and modern: he covers, inter alia, supersymmetry and Feynman path integrals in addition to the standard subjects.
The material in Faddeev and Yakubovskii is more standard, but in addition to that they have e.g. some nice bits of representation theory.
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*One quite advanced recent book on the subject is Quantum Theory, Groups and Representations by Peter Woit (see published version and final draft version on author's web site).


*Another good one is Mathematical Aspects of Quantum Field Theory, by de Faria and de Melo, published 2010, ISBN-13: 9780521115773.


*Mathematical Foundations of Quantum Mechanics - Mackey
A: *

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*E. Zeidler, Quantum Field theory I Basics in Mathematics and Physics (Springer, 2006), http://www.mis.mpg.de/ezeidler/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 mathematicians and physicists", as the subtitle says. It makes very interesting reading if you know already enough math and physics. You need a thorough knowledge of classical analysis, and some acquaintance with differential geometry and functional analysis. Apart from that, the book gives references to additional reading - 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, $H_\mathrm{int}=e^{-itH_0}He^{itH_0}$, is a PDO. Work on Wigner transforms is work on PDOs, etc..


*Other books using PDO, much more specialized:

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*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




*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

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*A. Neumaier and D. Westra, 
Classical and Quantum Mechanics via Lie algebras.



A: My three favorite resources on math foundations of QM haven't been mentioned yet, so I'll add them to the list.

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*Frederic Schuller's Lectures on Quantum Theory, which is actually a video lecture series (free on YouTube). If you're new to mathematical foundations of QM and think that linear algebra is a sufficient language for QM, this is the place to start (and these beliefs will be shattered).

*Valter Moretti's Spectral Theory and Quantum Mechanics. This could be the companion textbook to the above lecture series. It is great for beginners because it introduces (and proves) the basics of infinite-dimensional Hilbert space and relevant areas of functional analysis. It is very comprehensive, and also introduces some topics in mathematical physics with a thoroughness I have not seem elsewhere (e.g., quantum logic and quantum symmetries).

*Brian Hall's Quantum Theory for Mathematicians. This book assumes a bit more mathematical background than the previous one, but I find it to be "cleaner" mathematically.

