The book by Prof Gustafson and Prof Sigal has a good balance between mathematical rigor and not too much technicalities. Some results of more mathematical sofisticated nature are adressed to "Int to Spectral Theory. With Applications to Schrodinger Operators", by P. Hilsop
and I. Sigal.
In this way some of the main ideas of Quantum Mechanics are explained in this text in a clear way for someone with a mathematical
mind. It is apropriate for a person that does not know
well
the physics of Quantum Mechanics.
I believe any mathematician with some background in Functional Analysis and some basic knowledge of Partial Differential Equations and Fourier Analysis will enjoy this book.
There is some other books in the subject which contains all proofs,
all dificult technicalities, all mathematical details, but, in my opinion,
for a mathematician first lecture on the subject this the the one.

A. O. Lopes