As the subject of extensive research for over a century, spectral asymptotics for partial differential operators attracted the attention of many outstanding mathematicians and physicists. This book studies the eigenvalues of elliptic linear boundary value problems and has as its main content a collection of asymptotic formulas describing the distribution of eigenvalues with high sequential numbers. Asymptotic formulas are used to illustrate standard eigenvalue problems of mechanics and mathematical physics. The volume provides a basic introduction to all the necessary mathematical concepts and tools, such as microlocal analysis, billiards, symplectic geometry and Tauberian theorems. It is self-contained and would be suitable as a graduate text.
Main results Oscillatory integrals Construction of the wave group Singularities of the wave group Proof of main results Mechanical applications Appendix A. Spectral problem on the half-line Appendix B. Fourier Tauberian theorems Appendix C. Stationary phase formula Appendix D. Hamiltonian billiards: proofs Appendix E. Factorization of smooth functions and Taylor-type formulae References Principal notation Index.