Integration in function spaces and some of its applications

Description

These notes based on a series of lectures delivered at the Scuola Nor- male Superiore in May 1980 are meant to provide an introduction to the subject of functional integration. Although integration in function spaces owes its origins to Probability Theory on the one hand, and to Quantum Mechanics on the other, I assumed only most rudimentary familiarity with either of these disciplines. Also I did not attempt to write a minitextbook on the subject of integration in function spaces. I was much more concerned in exhibiting the spirit of the subject than in teaching it in an organized and systematic way. Consequently I stressed the formal and eschewed, perhaps even above and beyond the call of duty, technicalities. To borrow a famous saying, I came to Pisa to praise functional integration not to bury it. I hope the reader will keep this in mind and be willing to forgive me for being sketchy, incomplete and moving in too many directions at once. I also hope that the reader will understand that constraints the written word imposes on a speaker prevent him from including numerous side remarks of scientific and anecdotal nature which he could use freely to (hopefully!) edify (and perhaps even amuse) his live audience. CONTENTS page 1 Preface 5 Section 1 Introduction 7 Section 2 Construction of the Wiener measure and integration of some simple functional 23 Section 3 Elements of probabilistic potential theory 31 Section 4 Asymptotics of the number of bound states of certain Schrodinger equations and related topics 43 Section 5 Scattering length and capacity 49 Section 6 Feynman's approach to non-relativistic Quantum Mechanics 55 Section 7 Feynman's approach continued. Semi-classical Quantum Mechanics and a theorem of Morse 65 Section 8 Short time asymptotics and high eigenvalues of the Schrodinger equation 69 Section 9 Introduction to the Donsker-Varadhan long time asymptotics 77 Section 10 Introduction to the Donsker-Varadhan theory continued 81 References