1.3. OVERVIEW 7 Chapter 5: This part of the book introduces semiclassical defect measures and uses them to derive decay estimates for the damped wave equation (1.1.1), where a 0 on the flat torus Tn. A theorem of Rauch and Tay- lor provides a beautiful example of classical/quantum correspondence: the waves decay exponentially if all classical trajectories within a certain fixed time intersect the region where positive damping occurs. Chapter 6: In Chapter 6 we begin our study of the eigenvalue problem P (h)u(h) = E(h)u(h), for the operator P (h) := −h2Δ + V (x). We prove Weyl’s Law for the asymptotic distributions of eigenvalues as h 0, stating for all a b that #{E(h) | a E(h) b} = 1 (2πh)n (|{a |ξ|2 + V (x) b}| + o(1)) as h 0. Our proof is a semiclassical analogue of the classical Dirichlet– Neumann bracketing argument of Courant. Chapter 7: Chapter 7 deepens our study of eigenfunctions, first establish- ing an exponential vanishing theorem in the “classically forbidden” region. We derive as well a Carleman-type estimate: if u(h) is an eigenfunction of a Schr¨ odinger operator, then for any open set U ⊂⊂ Rn, u(h) L2(U) e−c/h u(h) L2(Rn) . This provides a quantitative estimate for quantum mechanical tunneling. We also present a self-contained “semiclassical” derivation of interior Schauder estimates for the Laplacian. Chapter 8: We return in Chapter 8 to the symbol calculus, first proving the semiclassical version of Beals’s Theorem, characterizing pseudodifferential operators. As an application we show how quantization commutes with exponentiation at the level of order functions and then use these insights to define useful generalized Sobolev spaces. This chapter also introduces wavefront sets and the notion of microlocality. Chapter 9: We next introduce the useful formalism of half-densities and use them to see how changing variables in a symbol affects the Weyl quan- tization. This motivates our introducing the new class of Kohn–Nirenberg symbols, which behave well under coordinate changes and are consequently useful later when we investigate the semiclassical calculus on manifolds. Chapter 10: Chapter 10 discusses the local construction of propagators, us- ing solutions of Hamilton–Jacobi PDE to build phase functions for Fourier
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