COURSE INFORMATION+ About the course:·Course Name: Semiclassical Microlocal Analysis.Instructor:Zuoqin Wang-Email: wangzuoq@ustc.edu.cn-Office:1601.LectureTime:Monday15:55pm-17:30pmWednesday19:30pm-21:05pm.Lecture Ro0m: 5304.Webpage:http:/staff.ustc.edu.cn/~wangzuog/Courses/20F-SMA/index.html[You may check thefollowingwebsiteformyold coursenotes:http:/staff.ustc.edu.cn/~wangzuog/Courses/14F-Semiclassical/SMA.htmlNote:The materials to be covered and the arrangement of topics in this semester will beslightly different from that one.].Homeworks: Will be assigned every two weeks or so.·Office Hours: Byappointment (via email)..Your grades: HWs + Essay Notes and Reference books:. Course Notes:Willbe uploaded to theCourse Webpageafter each lecture. Reference Books:The following are some nice reference books for this course.-SemiclassicalAnalysisbyM.Zworkski- Semi-Classical Analysis by V. Guillemin and S. Sternberg-Spectral Asymptotics in the Semi-Classical Limit by M.Dimassi and J. Sjostrand- An Introduction to Semi-classical and Microlocal Analysis by A. Martinez-Harmonic Analysis in Phase Space by G. Folland-TheAnalysis of Linear Partial Differential Operators IIIby L.Hormander1
Course Information ********************************************************************************* ♣ About the course: • Course Name: Semiclassical Microlocal Analysis • Instructor: Zuoqin Wang – Email: wangzuoq@ustc.edu.cn – Office: 1601 • Lecture Time: Monday 15:55 pm - 17:30 pm space Wednesday 19:30 pm - 21:05 pm • Lecture Room: 5304 • Webpage: http:/staff.ustc.edu.cn/∼wangzuoq/Courses/20F-SMA/index.html [You may check the following website for my old course notes: http:/staff.ustc.edu.cn/∼wangzuoq/Courses/14F-Semiclassical/SMA.html Note: The materials to be covered and the arrangement of topics in this semester will be slightly different from that one.] • Homeworks: Will be assigned every two weeks or so. • Office Hours: By appointment (via email). • Your grades: HWs + Essay ♦ Notes and Reference books: • Course Notes: Will be uploaded to the Course Webpage after each lecture. • Reference Books: The following are some nice reference books for this course. – Semiclassical Analysis by M. Zworkski – Semi-Classical Analysis by V. Guillemin and S. Sternberg – Spectral Asymptotics in the Semi-Classical Limit by M. Dimassi and J. Sjostrand ¨ – An Introduction to Semi-classical and Microlocal Analysis by A. Martinez – Harmonic Analysis in Phase Space by G. Folland – The Analysis of Linear Partial Differential Operators III by L. Hormander ¨ 1
LECTURE1—09/21/2020INTRODUCTION1.WHAT IS ANDWHY What is semiclassical microlocal analysis.Q: What is local?A:localize with respect to location in the space.-Configuration space:Euclidean space/manifolds- Analysis: PDE. Q: What is microlocal?A: Localize not only with respect to location in the space, but also with respect tocotangent directions at a given point.-Phase space: cotangent bundleAnalysis:Fourier/harmoic analysis. Q: What is classical?A:Classical mechanics (intheframework of Hamilton)-Symplectic geometry: Hamilton flow, Lagrangian submanifolds .. Q: What is quantum?A: Quantum mechanics (in the framework of Schrodinger)-Spectral theory: eigenvalues, eigenfunctions, ..Q: What is semiclassical?A: A theory/method describing the intermediate between classical mechanics andquantummechanics.Roughly speaking,"semiclassical microlocal analysis is an investigation of the math-ematical implications of the Bohr correspondence principle":a quantum system shallapproximate its classical model atformal h→O (or high frequency=eigenvalue →)limit.Why study?.Physics:provides themathematical theoryfor-classical-quantumcorrespondence-particle-wave correspondence-geometricoptics approximation. Mathematics: develops methods to-applyharmonic analysis and symplectic geometryto PDEs,-study asymptoticbehavior of eigenvalues/eigenfunctions-has broad applications tovarious subjects in mathematics including differen-tialgeometry,dynamicalsystem,numbertheoryetc2
LECTURE 1 — 09/21/2020 INTRODUCTION 1. What is and why ♥ What is semiclassical microlocal analysis • Q: What is local? A: localize with respect to location in the space. – Configuration space: Euclidean space/manifolds. – Analysis: PDE • Q: What is microlocal? A: Localize not only with respect to location in the space, but also with respect to cotangent directions at a given point. – Phase space: cotangent bundle – Analysis: Fourier/harmoic analysis • Q: What is classical? A: Classical mechanics (in the framework of Hamilton). – Symplectic geometry: Hamilton flow, Lagrangian submanifolds . • Q: What is quantum? A: Quantum mechanics (in the framework of Schrodinger). ¨ – Spectral theory: eigenvalues, eigenfunctions, . • Q: What is semiclassical? A: A theory/method describing the intermediate between classical mechanics and quantum mechanics. { Roughly speaking, “semiclassical microlocal analysis is an investigation of the mathematical implications of the Bohr correspondence principle”: a quantum system shall approximate its classical model at formal ~ → 0 (or high frequency = eigenvalue λ → ∞) limit. ♠ Why study? • Physics: provides the mathematical theory for – classical-quantum correspondence – particle-wave correspondence – geometric optics approximation • Mathematics: develops methods to – apply harmonic analysis and symplectic geometry to PDEs, – study asymptotic behavior of eigenvalues/eigenfunctions – has broad applications to various subjects in mathematics including differential geometry, dynamical system, number theory etc 2
3LECTURE1—09/21/2020INTRODUCTION2. WE WILL COVER...In this course weplan to cover+Backgrounds:.From physics: quantization and semiclassical limit-classical phase spacew symplectic- quantum operators on Hilbert space spectral theory-quantization: from classical description toquantum descriptionsemiclassical limit:fromquantumdescriptiontoclassicaldescription.From analysis: Fourier analysis, Method of stationary phase-Semiclassical Fourier transformEe-等p(x)dx.Fn() := (F)(JR- The lemma of stationary phase: The oscillatory integralel a(x)dx.In=has an asymptotic expansion as h→ O, with leading terma(pi)In = (2h)n/2 e1e sgn(d's(pa)Idet dp(P,)/2 + O($+1).de(pi)=0This formula will play a crucial role in studying semiclassical limit..Fromgeometry:Basic symplecticgeometrySymplectic manifolds:* Darboux: locally T*R"* A. Weinstein: I like to think of symplectic geometry as playing therole in mathematics of a language which can facilitate communicationbetween geometry and analysis*Hamiltonian mechanics-Lagrangian submanifolds*generating function:geometry analysis*canonical relations,symplectic“category"-(we assumetheknowledgeof basic theoryofmanifolds)-(In someapplicationsweassumetheknowledgeofbasicRiemanniangeom-etry)*The Laplace-Beltrami operatoras quantization ofl/?*geodesicflowas Hamiltonianflowassociatedtothefunctionllell
LECTURE 1 — 09/21/2020 INTRODUCTION 3 2. We will cover. In this course we plan to cover ♣ Backgrounds: • From physics: quantization and semiclassical limit – classical ! phase space ! symplectic – quantum ! operators on Hilbert space ! spectral theory – quantization: from classical description to quantum description – semiclassical limit: from quantum description to classical description • From analysis: Fourier analysis, Method of stationary phase – Semiclassical Fourier transform F~ϕ(ξ) := (F ϕ)( ξ ~ ) = Z Rn e − ix·ξ ~ ϕ(x)dx. – The lemma of stationary phase: The oscillatory integral I~ = Z Rn e i ϕ(x) ~ a(x)dx. has an asymptotic expansion as ~ → 0, with leading term I~ = (2π~) n/2 X dϕ(pi)=0 e i ϕ(pi ) ~ e iπ 4 sgn(d 2ϕ(pi)) a(pi) | det d 2ϕ(pi)| 1/2 + O(~ n 2 +1 ). This formula will play a crucial role in studying semiclassical limit. • From geometry: Basic symplectic geometry – Symplectic manifolds: ∗ Darboux: locally T ∗R n ∗ A. Weinstein: I like to think of symplectic geometry as playing the role in mathematics of a language which can facilitate communication between geometry and analysis ∗ Hamiltonian mechanics – Lagrangian submanifolds ∗ generating function: geometry ! analysis ∗ canonical relations, symplectic “category” – (we assume the knowledge of basic theory of manifolds) – (In some applications we assume the knowledge of basic Riemannian geometry) ∗ The Laplace-Beltrami operator ∆ as quantization of kξk 2 ∗ geodesic flow as Hamiltonian flow associated to the function kξk
4LECTURE1—09/21/2020INTRODUCTION Semiclassical pseudodifferential operators (as quantizations of symbols).We can quantize the position function xand the momentum functionkXkwQk=multiplicatonbyxkhaSkWPk=V-iaxkand thus can quantize the energyfunction to the Schrodinger operator1e12+V()=_H=-△+ V(x).22But how to quantizemore general functions like xii? Note:xisi = sixi but Qio Pi+Pr oQ1!Answer:usesemiclassicalFouriertransformSymbols and various semiclassicalquantizations-Forafunctionawhichliesinsomeniceclass(symbols)1oa'(x, hD)(p)(x) :a(x,)o(y)dyd(2元h)" Jg*1aR(x, hD)()(x) :a(y,s)p(y)dye(2元h)n1x+ya"(x,D)(0)(x) =$)p(y)dyde(2元h)n2For example, if a(x,)= xisi,thena'=QiPi, dR=PiQi, aW_ QiPI+PiQl2-composition law: the composition of semiclassical PsDOs is still a semiclas-sicalPsDOwithcomputablesymbols.Inparticular,[a"(x,hD),bW(x,hD)| ="(a, b)"(x,hD) +O(3).Properties of PsDOs as operators-L-boundedness,compactness- Ellipticity, paramatrix (pseudo-inverse)-generalized Sobolev space-wavefrontset.Global theory-Invariance under coordinate change-PsDOs on manifolds
4 LECTURE 1 — 09/21/2020 INTRODUCTION ♦ Semiclassical pseudodifferential operators (as quantizations of symbols) • We can quantize the position function xk and the momentum function ξk xk Qk = multiplicaton by xk ξk Pk = ~ √ −1 ∂ ∂xk and thus can quantize the energy function to the Schrodinger operator ¨ H = |ξ| 2 2 + V(x) Hˆ = − ~ 2 2 ∆ + V(x). But how to quantize more general functions like x1ξ1? Note: x1ξ1 = ξ1 x1 but Q1 ◦ P1 , P1 ◦ Q1! Answer: use semiclassical Fourier transform • Symbols and various semiclassical quantizations – For a function a which lies in some nice class (symbols), a L (x, ~D)(ϕ)(x) = 1 (2π~) n Z Rn Z Rn e i (x−y)·ξ ~ a(x, ξ)ϕ(y)dydξ, a R (x, ~D)(ϕ)(x) = 1 (2π~) n Z Rn Z Rn e i (x−y)·ξ ~ a(y, ξ)ϕ(y)dydξ, a W(x, ~D)(ϕ)(x) = 1 (2π~) n Z Rn Z Rn e i (x−y)·ξ ~ a( x + y 2 , ξ)ϕ(y)dydξ. For example, if a(x, ξ) = x1ξ1, then a L = Q1P1, a R = P1Q1, a W = Q1P1 + P1Q1 2 . – composition law: the composition of semiclassical PsDOs is still a semiclassical PsDO with computable symbols. In particular, h a W(x, ~D), b W(x, ~D) i = ~ i {a, b} W(x, ~D) + O(~ 3 ), • Properties of PsDOs as operators – L 2 -boundedness, compactness – Ellipticity, paramatrix (pseudo-inverse) – generalized Sobolev space – wave front set • Global theory – Invariance under coordinate change – PsDOs on manifolds
LECTURE1—09/21/2020INTRODUCTION5 Semi-classical Fourier integral operators (as quantizations of canonical relations).First example:thepropagatorU(t) = e-io/n,is a FIO that quantize the Hamilton flow pr of q(x, ):Theorem 2.1 (Egorov's theorem).Let b,(x,)=pa be the“classical flow-out"of the symbol a(x,) along the Hamiltonian fow p, of q(x, ). Theneir0/ha"(x,hD)e-ir0/n = bW(x, hD) + O(h)..Oscillatory half densities associated with canonical relations.semiclassicalFIO,symboliccalculusApplications to spectral theoryIn spectral theory we study eigenvalues/eigenfunction of operators that arise fromanalysis, geometry, physics etc. By definition they are non-trivial solutions toPu= Nu.where M is a real number called an eigenvalue of P, and u is called an eigenfunctionassociated to . In most interesting cases, the operator Pis an elliptic (pseudo)differentialoperator,andtheeigenvaluesformanincreasingdiscretesequenceM1 ≤ N2 ≤ 3 ≤ ...180and the associated eigenfunctions can be chosen to form a normalized basis of the corresponding Hilbert space..ThedistributionofeigenvaluesFor example, consider the eigenvalue counting functionN() = #(jI 2, ≤),WewillproveTheorem 2.2 (Weyl law). For the Schrodinger operator H (or for elliptic semi-classicalpseudodifferentialoperatorssatisfyingspecificproperties)N() =Vol(((x, ) I H(x, ) <) + O(h))(2元h)n·The spatial (in configuration space as well as in phase space)distribution of eigen-functionsFor example,we will proveTheorem2.3 (Quantum ergodicity theorem). Suppose (M,g)is a Riemannianmanifold whose geodesic flow is ergodic.Then there exists a density one sequenceof L2-normalized Laplacian eigenfunctions uj such that for any f,luiPf(x)dx- + f(x)dx
LECTURE 1 — 09/21/2020 INTRODUCTION 5 ♥ Semi-classical Fourier integral operators (as quantizations of canonical relations) • First example: the propagator U(t) = e −itQ/~ , is a FIO that quantize the Hamilton flow ρt of q(x, ξ): Theorem 2.1 (Egorov’s theorem). Let bt(x, ξ) = ρ ∗ t a be the “classical flow-out” of the symbol a(x, ξ) along the Hamiltonian flow ρt of q(x, ξ). Then e itQ/~ a W(x, ~D)e −itQ/~ = b W t (x, ~D) + O(~). • Oscillatory half densities associated with canonical relations • semiclassical FIO, symbolic calculus ♠ Applications to spectral theory In spectral theory we study eigenvalues/eigenfunction of operators that arise from analysis, geometry, physics etc. By definition they are non-trivial solutions to Pu = λu, where λ is a real number called an eigenvalue of P, and u is called an eigenfunction associated to λ. In most interesting cases, the operator P is an elliptic (pseudo)differential operator, and the eigenvalues form an increasing discrete sequence λ1 ≤ λ2 ≤ λ3 ≤ · · · → ∞ and the associated eigenfunctions can be chosen to form a normalized basis of the corresponding Hilbert space. • The distribution of eigenvalues For example, consider the eigenvalue counting function N(λ) = #{j | λj ≤ λ}. We will prove Theorem 2.2 (Weyl law). For the Schrodinger operator H (or for elliptic semi- ˆ classical pseudodifferential operators satisfying specific properties) N(λ) = 1 (2π~) n Vol({(x, ξ) | H(x, ξ) < λ} + O(~)). • The spatial (in configuration space as well as in phase space) distribution of eigenfunctions For example, we will prove Theorem 2.3 (Quantum ergodicity theorem). Suppose (M, g) is a Riemannian manifold whose geodesic flow is ergodic. Then there exists a density one sequence of L2 -normalized Laplacian eigenfunctions ujk such that for any f , Z M |ujk | 2 f(x)dx → ? M f(x)dx