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Semi-classical analysis. (English) Zbl 1298.58001
Somerville, MA: International Press (ISBN 978-1-57146-276-3/pbk). xxiv, 446 p. (2013).
For the (positive) Laplace-Beltrami operator \(\Delta_g\) on a compact Riemannian manifold \((M,g)\), Weyl’s law states that the number of eigenvalues below \(\lambda\) is asymptotic, as \(\lambda\to+\infty\), to \((2\pi)^{-\dim M}\) times the Liouville measure of \(\{(x,\xi)\in T^* M; |\xi|^2<\lambda\}\). Additional information about the spectral asymptotics is obtained from the wave trace, which is the distributional trace of the unitary wave group, \(\exp(it\sqrt{\Delta_g})\). The singularities of the wave trace are located in the periods of the closed geodesics. The seminal L. Hörmander [Acta Math. 127, 79–183 (1971; Zbl 0212.46601)] introduced the calculus of Fourier Integral Operators (FIO), which is the basic tool for proving these results. The wave group is a FIO, and the wavefront set of its Schwartz kernel \(U(t,x,y)\) is propagated along the geodesic flow. The wave trace \(\int_M U(t,x,x)d_gx\) is defined and evaluated by writing it as the image of \(U\) under the composition of two FIOs, the pullback to the diagonal followed by the pushforward along the diagonal.
In its original polyhomogeneous form the FIO calculus of microlocal analysis is designed to deal with regularity questions for linear differential equations. The polyhomogeneous setting is not adequate however for an asymptotic spectral analysis of the Schrödinger operator \(h^2\Delta_g + V(x)\). In view of the Bohr correspondence principle the (principal) symbol should be \(|\xi|^2+V(x)\) not \(h^2|\xi|^2\), which is the homogeneous principal symbol. Semi-classical analysis is a variant of microlocal analysis suitable for stating and proving assertions in spectral asymptotics. It has been successfully applied to many problems, not only for the Schrödinger operator where the asymptotic parameter \(h\downarrow 0\) is Planck’s constant. (The numerical value of Planck’s constant is not a constant but depends on chosen physical units; therefore letting it tend to zero is appropriate when moving from subatomic towards macroscopic scales.)
We describe the contents of the book under review. The first part, Chapters 2 to 7, deals with symplectic geometry. For the Schrödinger operator this is the geometry of the classical mechanics limit. The authors adopt a categorical approach with symplectic manifolds as objects and canonical relations as morphisms. This symplectic “category” is not quite a category however, because the composition of morphisms is defined only under the additional assumption of clean intersection. Canonical relations between cotangent bundles \(T^*X_1\) and \(T^*X_2\) are Lagrangian submanifolds of the symplectic manifold \((T^*X_1^-\times T^*X_2,\omega_2-\omega_1)\), where \(\omega_j\) is the symplectic form of \(T^*X_j\). Enlarging the symplectic “category” with point objects, every Lagrangian manifold becomes also a canonical relation. A Lagrangian submanifold \(\Lambda\) is called exact when it is equipped with a smooth function \(\psi:\Lambda\to\mathbb{R}\), the phase of \(\Lambda\), such that \(d\psi\) equals the restriction of the canonical one-form to \(\Lambda\). The prototypical Lagrangian submanifolds are the horizontal ones, i.e., the images \(\Lambda_\varphi\) of differentials \(d\varphi\), regarded as sections of \(T^*X\), of smooth generating functions \(\varphi:X\to\mathbb{R}\). Moreover, \(\Lambda_\varphi\) is exact with phase equal to \(\varphi\) viewed as a function on \(\Lambda_\varphi\). Generally, (exact) Lagrangian manifolds are locally representable using fibrations and generating functions. Canonical relations are equipped with their half-density and Maslov bundles. These bundles behave functorially under clean compositions, thereby creating an enhancement of the symplectic category.
The second part, Chapters 8 to 12, is the core of the book. Here a semi-classical calculus is developed and applied. The space \(I(X,\Lambda)\) of oscillatory half-densities associated with an exact Lagrangian submanifold \((\Lambda,\psi)\) of \( T^*X\) consists of \(h\)-dependent distributions on \(X\). Locally, the elements of \(I(X,\Lambda)\) are push-forwards by fibrations \(\pi:Z\to X\) of oscillatory half-densities \(h^\kappa a(z,h)e^{i\varphi(z)/h}\). The generating function \(\varphi\in C^\infty(Z)\) satisfies \(\Lambda=\Gamma_\pi\circ\Lambda_\varphi\), where \(\Gamma_\pi\) is the canonical relation induced by \(\pi\). Moreover, the additive constant in \(\varphi\) is fixed by demanding that \(\psi\) and \(\varphi\) are equal if both functions are regarded (via the obvious pullbacks) as functions on \(\Lambda_\varphi\). This requirement, which is not needed in the classical theory where \(\varphi\) is homogeneous, is important for the semi-classical symbol calculus of the book. The symbol space \(I^k(X,\Lambda)/I^{k+1}(X,\Lambda)\) is identified with the space of smooth sections of a line bundle \(\mathbb{L}\to\Lambda\), which is the tensor product of the Maslov bundle with the half-density bundle. Semi-classical FIOs are \(h\)-dependent operators having oscillatory half-densities as their Schwartz kernels. Semi-classical pseudo-differential operators \(A=a(x,hD,h)\in \Psi(X)\) are semi-classical FIOs with the underlying canonical relation induced by the identity map. This pattern is familiar from Hörmander’s paper cited above, and it is one goal of the book to extend it from the classical polyhomogeneous to the semi-classical case. Classical pseudo-differential operators with polyhomogeneous symbol are characterized as follows: \(A\) is such an operator if and only if \(AP\in \Psi_{00}(X)\) for every \(P\in\Psi_{00}(X)\). Here \(\Psi_{00}(X)\) denotes the space of semi-classical pseudo-differential operators with microsupport compact and disjoint from the zero section in \(T^*X\). A calculus of semi-classical pseudodifferential operators \(\Psi^k S^m(X)\) is developed, where \(k\) refers to the degree in \(h\) and \(m\) to the order as an operator. The calculus also includes a modulo \(h^\infty\) functional calculus for selfadjoint elliptic operators. Among the applications presented are spectral invariants of the Schrödinger operator and Gutzwiller’s trace formula. In chapter 12 the exactness assumption on Lagrangian submanifolds \(\Lambda\) is relaxed to integrality. This means that there is a \(C^\infty\) map \(f:\Lambda\to S^1\) such that the restriction of the canonical one-form to \(\Lambda\) equals \((2\pi i f)^{-1} df\). When the asymptotic parameter \(h\) is allowed to range in reciprocals of integers only, then semi-classical calculus extends to the integral symplectic “category”. This allows the authors to discuss aspects of geometric quantization and of the representation theory of compact Lie groups \(G\) in their framework. To an integral coadjoint orbit \(O\), which corresponds to an irreducible representation, is attached an integral Lagrangian manifold \(\Lambda_O \subset T^*G\). The Weyl and the Kirillov character formulas are related to symbols of particular elements of \(I^0(G,\Lambda_O)\), actually sequences of characters, and to generating functions of \(\Lambda_O\). In addition, equivariant spectral problems are treated in the integral setting of semi-classical analysis.
The third part of the book, Chapters 13 to 16, consists essentially of appendices on functional analysis, on differential calculus on manifolds, on the method of stationary phase, and on the Weyl transform, which is discussed in the context of the Stone-von-Neumann theorem.
There exist several good books on semi-classical analysis, notably by Robert, Helffer, Dimassi-Sjöstrand, Ivrii, Martinez, and Zworski. The book under review must be added to this list. A distinctive feature is that the authors strive to get the global aspects of semi-classical geometry and analysis right. However, for a fuller picture and for more applications of semi-classical analysis a reader should also consult at least one of the other above-mentioned books. The style of exposition is oral. In particular, previous material is often reviewed with a new perspective, thus helping the reader. On the other hand, arguments are sometimes sketchy, and occasionally even questionable. Still, a reader may benefit greatly from exploring this book.

MSC:
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58J40 Pseudodifferential and Fourier integral operators on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35S99 Pseudodifferential operators and other generalizations of partial differential operators
53D05 Symplectic manifolds (general theory)
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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