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Spectral Theory and Applications. an Elementary Introductory Course

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Spectral theory and applications. An elementary introductory course. Bucarest Version 2010 Bernard Helffer Universit´eParis-Sud, D´epartement de Math´ematiques, UMR 8628 du CNRS, Bat. 425, F-91405 Orsay Cedex, FRANCE March 26, 2010 Abstract We intend to present in this course the basic tools in spectral analy- sis and to illustrate the theory by presenting examples coming from the Schr¨odinger operator theory and from various branches of physics : sta- tistical mechanics, superconductivity, fluid mechanics. We also give some introduction to non self-adjoint operators theory with emphasis on the role of the pseudo-spectrum. Other examples are treated in the 2008- 2009 version. Contents 1 Introduction 4 1.1 The free Laplacian . 4 1.2 The harmonic oscillator . 4 1.3 The problem of the boundary . 6 1.3.1 Ill-posed problems . 6 1.3.2 The periodic problem . 7 1.3.3 The Dirichlet problem . 7 1.3.4 The Neumann problem . 7 1.3.5 Conclusion . 8 2 Unbounded operators, adjoints, Selfadjoint operators 8 2.1 Unbounded operators . 8 2.2 Adjoints. 11 2.3 Symmetric and selfadjoint operators. 14 1 3 Representation theorems 16 3.1 Riesz's Theorem. 16 3.2 Lax-Milgram's situation. 16 3.3 An alternative point of view: V; H;V 0. 17 4 Semi-bounded operators and Friedrichs extension. 20 4.1 Definition . 20 4.2 Analysis of the Coulomb case. 21 4.3 Friedrichs's extension . 24 4.4 Applications . 25 5 Compact operators : general properties and examples. 30 5.1 Definition and properties. 30 5.2 Examples . 30 5.3 Adjoints and compact operators . 33 5.4 Precompactness . 34 6 Spectral theory for bounded operators. 36 6.1 Fredholm's alternative . 36 6.2 Resolvent set for bounded operators . 37 6.3 Spectral theory for compact operators . 39 6.4 Spectrum of selfadjoint operators. 41 6.5 Spectral theory for compact selfadjoint operators . 42 7 Examples. 44 7.1 The transfer operator. 44 7.1.1 Compactness . 44 7.1.2 About the physical origin of the problem. 44 7.1.3 Krein-Rutman's Theorem. 48 7.2 The Dirichlet realization, a model of operator with compact re- solvent. 50 7.3 Extension to operators with compact resolvent . 54 7.4 Operators with compact resolvent : the Schr¨odingeroperator in an unbounded domain. 54 7.5 The Schr¨odingeroperator with magnetic field . 56 7.6 Laplace Beltrami operators on a Riemannian compact manifold 58 8 Selfadjoint unbounded operators and spectral theory. 60 8.1 Introduction . 60 8.2 Spectrum. 63 8.3 Spectral family and resolution of the identity. 64 8.4 The spectral decomposition Theorem. 69 8.5 Applications of the spectral theorem: . 73 8.6 Examples of functions of a selfadjoint operator . 74 8.7 Spectrum and spectral measures . 75 2 9 Non-self adjoint operators and -pseudospectrum 77 9.1 Main definitions and properties . 77 9.2 -Pseudospectrum : complete analysis of the differentiation op- erator. 79 9.3 Another example of non selfadjoint operator without spectrum . 83 9.4 The non selfadjoint harmonic oscillator . 87 9.5 The complex Airy operator in R . 88 10 Essentially selfadjoint operators 91 10.1 Introduction . 91 10.2 Basic criteria. 92 10.3 The Kato -Rellich theorem . 94 10.4 Other criteria of selfadjointness for Schr¨odingeroperators . 96 11 Non-selfadjoint case : Maximal accretivity and application to the Fokker-Planck operator 99 11.1 Accretive operators . 99 11.2 Application to the Fokker-Planck operator . 99 11.3 Decay of the semi-group and -pseudospectra . 101 11.4 Application : The complex Airy operator in R+ . 102 12 Discrete spectrum, essential spectrum 105 12.1 Discrete spectrum . 105 12.2 Essential spectrum . 105 12.3 Basic examples: . 106 12.4 On the Schr¨odingerOperator with constant magnetic field . 106 12.4.1 Dimension 2 . 106 12.4.2 The case of R2 . 107 12.4.3 Magnetic Schr¨odingeroperators in dimension 3 . 108 12.5 Weyl's criterion: . 108 13 The max-min principle 112 13.1 Introduction . 112 13.2 On positivity . 112 13.3 Variational characterization of the discrete spectrum . 113 13.4 Max-min principle . 115 13.5 CLR inequality . 119 13.6 Essential spectrum and Persson's Theorem . 120 14 Exercises and Problems 122 14.1 Exercises . 122 14.2 Problems . 128 3 1 Introduction Our starting point could be the theory of Hermitian matrices, that is of the matrices satisfying : A? = A. If we look for eigenvectors and corresponding eigenvalues of A, that is for pairs (u; λ) with u 2 Ck, u 6= 0 and λ 2 C such that Au = λu, we know that the eigenvalues are real and that one can find an orthonormal basis of eigenvectors associated with real eigenvalues. In order to extend this theory to the case of spaces with infinite dimension (that is replacing the space Cm by a general Hilbert space H), the first attempt consists in developing the theory of compact selfadjoint operators. But it is far to cover all the interesting cases that are present in Quantum Mechanics. So our aim is to present a general theory but it is perhaps good to start by looking at specific operators and to ask naive questions about the existence of pairs (u; λ) with u in some suitable domain, u 6= 0 and λ 2 C such that Au = λu. We shall discover in particular that the answer at these questions may depend strongly on the choice of the domain and on the precise definition of the operator. 1.1 The free Laplacian The Laplacian −∆ has no eigenfunctions in L2, but it has for any λ 2 R+ an eigenfunction in S0(Rm) (actually in L1) and for any λ 2 C an eigenfunction in D0(Rm). So what is the right way to extend the theory of Hermitian matrices on Ck ? On the other hand, it is easy to produce (take for simplicity m = 1) approximate 2 p1 x−n eigenfunctions in the form un(x) = n exp ix·ξ χ( n ), where χ is a compactly supported function of L2-norm equal to 1. 1.2 The harmonic oscillator As we shall see the harmonic oscillator H = −d2=dx2 + x2 plays a central role in the theory of quantum mechanics. When looking for eigenfunctions in S(R), we obtain that there is a sequence of eigenvalues λn (n 2 N) λn = (2n − 1) : In particular the fundamental level (in other words the lowest eigenvalue) is λ1 = 1 and the splitting between the two first eigenvalues is 2. The first eigenfunction is given by x2 φ (x) = c exp − (1.2.1) 1 1 2 4 and the other eigenfunctions are obtained by applying the creation operator L+ = −d=dx + x : (1.2.2) We observe indeed that H = L+ · L− + 1 ; (1.2.3) where L− = d=dx + x ; (1.2.4) and has the property − L φ1 = 0: (1.2.5) Note that if u 2 L2 is a distributional solution of L+u = 0, then u = 0. Note 2 − also that if u 2 L is a distributional solution of L u = 0, then u = µφ1 for some µ 2 R. The nth-eigenfunction is then given by + n−1 φn = cn(L ) φ1 : This can be shown by recursion using the identity L+(H + 2) = HL+ : (1.2.6) x2 It is easy to see that φn(x) = Pn(x) exp − 2 where Pn(x) is a polynomial of order n − 1. One can also show that the φn are mutually orthogonal. The proof of this point is identical to the finite dimensional case, if we observe the following identity (expressing that H is symmetric) : < Hu; v >L2 =< u; Hv >L2 ; 8u 2 S(R); 8v 2 S(R) ; (1.2.7) which is obtained through an integration by parts. Then it is a standard exercise to show that the family (φn)n2N is total and that we have obtained an orthonormal hilbertian basis of L2, which in some sense permits to diagonalize the operator H. Another way to understand the completeness is to show that starting of an eigen- function u in S0(R) associated with λ 2 R solution (in the sense of distribution) of Hu = λu ; − k then there exists k 2 N and ck 6= 0 such that (L ) u = ckφ1 and that the corresponding λ is equal to (2k + 1). For this proof, we admit that any eigenfunction can be shown to be in S(R) and use the identity L−(H − 2) = HL− ; (1.2.8) and the inequality < Hu; u >≥ 0 ; 8u 2 S(R) : (1.2.9) 5 This property is called "positivity" of the operator. Actually one can show by various ways that 2 < Hu; u >≥ jjujj ; 8u 2 S(R) : (1.2.10) One way is to first establish the Heisenberg Principle1 : 2 0 jjujj 2 ≤ 2jjxujj 2 jju jj 2 ; 8u 2 S( ) : (1.2.11) L (R) L L R The trick is to observe the identity d d 1 = · x − x · : (1.2.12) dx dx The inequality (1.2.10) is simply the consequence of the identity < Hu; u >= jju0jj2 + jjxujj2 ; (1.2.13) which is proved through an integration by parts, and of the application in (1.2.11) of Cauchy-Schwarz inequality. Another way is to directly observe the identity < Hu; u >= jjL−ujj2 + jjujj2 ; 8u 2 S : (1.2.14) 1.3 The problem of the boundary d2 We mainly consider the operator − dx2 and look at various problems that can be asked naively about the existence of eigenfunctions for the problem in L2(]0; 1[).
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