The spectral theory of linear operators in Hilbert spaces is the most
important tool in the mathematical formulation of quantum mechanics; in
fact, linear ope- tors and quantum mechanics have had a symbiotic
relationship. However, typical physicstextbooks on quantum mechanics
givejust a roughsketch of operator t- ory, occasionallytreating linear
operatorsas matricesin ?nite-dimensional spaces; the implicit
justi?cation is that the details of the theory of unbounded operators
are involved and those texts are most interested in applications.
Further, it is also assumed that mathematical intricacies do not show up
in the models to be d- cussedorareskippedby"heuristicarguments.
"Inmanyoccasionssomequestions, such as the very de?nition of the
hamiltonian domain, are not touched, leaving an open door for
controversies, ambiguities and choices guided by personal tastes and ad
hoc prescriptions. All in all, sometimes a blank is left in the
mathematical background of people interested in nonrelativistic quantum
mechanics. Quantum mechanics was the most profound revolution in
physics; it is not natural to our common sense (check, for instance, the
wave-particle duality) and the mathematics may become crucial when
intuition fails. Even some very simple
systemspresentnontrivialquestionswhoseanswersneedamathematicalapproach.
For example, the Hamiltonian of a quantum particle con?ned to a box
involves a choice of boundary conditions at the box ends; since di?erent
choices imply di?erentphysicalmodels,
studentsshouldbeawareofthebasicdi?cultiesintrinsic
tothis(inprinciple)verysimple model,
aswellasinmoresophisticatedsituations. The theory of linear operators
and their spectra constitute a wide ?eld and it is expected that the
selection of topics in this book will help to ?ll this theoretical gap.
Ofcoursethisselectionisgreatlybiasedtowardthepreferencesofthe author.
