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Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

Author: Johannes Sjöstrand

Publisher: Springer International Publishing

Book Series : Pseudo-Differential Operators

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

About this book

The asymptotic distribution of eigenvalues of self-adjoint differential operators in the high-energy limit, or the semi-classical limit, is a classical subject going back to H. Weyl of more than a century ago.

In the last decades there has been a renewed interest in non-self-adjoint differential operators which have many subtle properties such as instability under small perturbations. Quite remarkably, when adding small random perturbations to such operators, the eigenvalues tend to distribute according to Weyl's law (quite differently from the distribution for the unperturbed operators in analytic cases). A first result in this direction was obtained by M. Hager in her thesis of 2005. Since then, further general results have been obtained, which are the main subject of the present book.

Additional themes from the theory of non-self-adjoint operators are also treated. The methods are very much based on microlocal analysis and especially on pseudodifferential operators. The reader will find a broad field with plenty of open problems.

Frontmatter

1. Introduction

Abstract

Non-self-adjoint operators is an old, sophisticated and highly developed subject. See for instance Carleman for an early result on Weyl type asymptotics for the real parts of the large eigenvalues of operators that are close to self-adjoint ones, with later results by Markus and Matseev in the same direction. (See the classical works Weyl, Avakumović , Hörmander for the asymptotics of large eigenvalues of elliptic self-adjoint operators and Robert and Dimassi and Sjöstrand for corresponding results in the semi-classical case, not to mention numerous deep and sophisticated results by Ivrii and others.) Abstract theory with the machinery of s-numbers can be found in the book of Gohberg and Krein. Other quite classical results concern upper bounds on the number of eigenvalues in various regions of the complex plane and questions about completeness of the set of all generalized eigenvectors.

Johannes Sjöstrand

Basic Notions, Differential Operators in One Dimension

Frontmatter

2. Spectrum and Pseudo-Spectrum

Abstract

In this book all Hilbert spaces will be assumed to separable for simplicity. In this section we review some basic definitions and properties; we refer to Kato (Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer, New York, 1966), Reed and Simon (Methods of modern mathematical physics. I. Functional analysis, 2nd edn. Academic, New York, 1980; Methods of modern mathematical physics. II. Fourier analysis, self adjointness. Academic, New York, 1975; Methods of modern mathematical physics. IV. Analysis of operators. Academic, New York, 1978), Riesz and Sz.-Nagy (Leçons d’analyse fonctionnelle, Quatrième édition. Académie des Sciences de Hongrie, Gauthier-Villars, Editeur- Imprimeur-Libraire, Paris; Akadémiai Kiadó, Budapest 1965) for more substantial presentations.

Johannes Sjöstrand

3. Weyl Asymptotics and Random Perturbations in a One-Dimensional Semi-classical Case

Abstract

We consider a simple model operator P in dimension 1 and show how random perturbations give rise to Weyl asymptotics in the interior of the range of P. We follow rather closely the work of Hager (Ann Henri Poincaré 7(6):1035–1064, 2006) with some input also from Bordeaux Montrieux (Loi de Weyl presque sûreet résolvante pour des opérateurs différentiels nonautoadjoints, thèse, CMLS, Ecole Polytechnique, 2008) and Hager–Sjöstrand (Math Ann 342(1):177–243, 2008). Some of the general ideas appear perhaps more clearly in this special situation.

Johannes Sjöstrand

4. Quasi-Modes and Spectral Instability in One Dimension

Abstract

In this section we describe the general WKB construction of approximate “asymptotic” solutions to the ordinary differential equation

$$\displaystyle P(x,hD_x)u=\sum _{k=0}^m b_k(x)(hD_x)^ku=0, $$

on an interval α < x < β, where we assume that the coefficients b _k ∈ C ^∞(]α, β[). Here h ∈ ]0, h ₀] is a small parameter and we wish to solve (above equation) up to any power of h. We look for u in the form

$$\displaystyle u(x;h)=a(x;h)e^{i\phi (x)/h}, $$

where ϕ ∈ C ^∞(]α, β[) is independent of h. The exponential factor describes the oscillations of u, and when ϕ is complex valued it also describes the exponential growth or decay; a(x;h) is the amplitude and should be of the form

$$\displaystyle a(x;h)\sim \sum _{\nu =0}^\infty a_\nu (x)h^\nu \mbox{ in }C^\infty (]\alpha ,\beta [). $$

Johannes Sjöstrand

5. Spectral Asymptotics for More General Operators in One Dimension

Abstract

In this chapter, we generalize the results of Chap. 3. The results and the main ideas are close, but not identical, to the ones of Hager (Ann Henri Poincaré 7(6):1035–1064, 2006). We will use some h-pseudodifferential machinery, see for instance Dimassi and Sjöstrand (Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series, vol 268. Cambridge University Press, Cambridge, 1999).

Johannes Sjöstrand

6. Resolvent Estimates Near the Boundary of the Range of the Symbol

Abstract

The purpose of this chapter is to give quite explicit bounds on the resolvent near the boundary of Σ(p) (or more generally, near certain “generic boundary-like” points.) The result is due (up to a small generalization) to Montrieux (Estimation de résolvante et construction de quasimode pres du bord du pseudospectre, 2013) and improves earlier results by Martinet (Sur les propriétés spectrales d’opérateurs nonautoadjoints provenant de la mécanique des fluides, 2009) about upper and lower bounds for the norm of the resolvent of the complex Airy operator, which has empty spectrum (Almog, SIAM J Math Anal 40:824–850, 2008). There are more results about upper bounds, and some of them will be recalled in Chap. 10 when dealing with such bounds in arbitrary dimension.

Johannes Sjöstrand

7. The Complex WKB Method

Abstract

In this chapter we shall study the exponential growth and asymptotic expansions of exact solutions of second-order differential equations in the semi-classical limit. As an application, we establish a Bohr-Sommerfeld quantization condition for Schrödinger operators with real-analytic complex-valued potentials.

Johannes Sjöstrand

8. Review of Classical Non-self-adjoint Spectral Theory

Abstract

The first section of this chapter deals with Fredholm theory in the spirit of Appendix A in Helffer and Sjöstrand (Mm Soc Math Fr (NS) 24–25:1–228, 1986), see also an appendix in Melin and Sjöstrand (Astérique 284:181–244, 2003) and Sjöstrand and Zworski (Ann Inst Fourier 57:2095–2141, 2007). The remaining sections give a brief account of the very beautiful classical theory of non-self-adjoint operators, taken from a section in Sjöstrand (Lectures on Resonances) which is a brief account of parts of the classical book by Gohberg and Krein (Introduction to the Theory of Linear Non-Selfadjoint Operators. Translations of Mathematical Monographs, vol 18. AMS, Providence, 1969).

Johannes Sjöstrand

Some general results

Frontmatter

9. Quasi-Modes in Higher Dimension

Abstract

Recall that if a(x, ξ) and b(x, ξ) are two C ¹-functions defined on some domain in ${\mathbf {R}}^{2n}_{x,\xi }$, then we can define the Poisson bracket to be the C ⁰-function on the same domain given by

$$\displaystyle \{ a,b\} =a^{\prime }_\xi \cdot b^{\prime }_x-a^{\prime }_x \cdot b^{\prime }_\xi =H_a(b). $$

Here $H_a=a^{\prime }_\xi \cdot \partial _x-a^{\prime }_x\cdot \partial _\xi $ denotes the Hamilton vector field of a. The following result is due to Zworski, who obtained it via a semi-classical reduction from the above mentioned result of Hörmander. A direct proof was given in Dencker et al. and here we give a variant. We will assume some familiarity with symplectic geometry.

Johannes Sjöstrand

10. Resolvent Estimates Near the Boundary of the Range of the Symbol

Abstract

In this chapter, which closely follows, we study bounds on the resolvent of a non-self-adjoint h-pseudodifferential operator P with (semi-classical) principal symbol p when h → 0, when the spectral parameter is in a neighborhood of certain points on the boundary of the range of p. In Chap. 6 we have already described a very precise result of W. Bordeaux Montrieux in dimension 1. Here we consider a more general situation; the dimension can be arbitrary and we allow for more degenerate behaviour. The results will not be quite as precise as in the one-dimensional case.

Johannes Sjöstrand

11. From Resolvent Estimates to Semigroup Bounds

Abstract

In Chap. 10 we saw a concrete example of how to get resolvent bounds from semigroup bounds. Naturally, one can go in the opposite direction and in this chapter we discuss some abstract results of that type, including the Hille–Yoshida and Gearhardt–Prüss–Hwang–Greiner theorems. As for the latter, we also give a result of Helffer and the author that provides a more precise bound on the semigroup.

Johannes Sjöstrand

12. Counting Zeros of Holomorphic Functions

Abstract

In this chapter we will generalize Proposition 3.4.6 of Hager about counting the zeros of holomorphic functions of exponential growth. In Hager and Sjöstrand (Math Ann 342(1):177–243, 2008. http://arxiv.org/abs/math/0601381) we obtained such a generalization, by weakening the regularity assumptions on the functions ϕ. However, due to some logarithmic losses, we were not quite able to recover Hager’s original result, and we still had a fixed domain Γ with smooth boundary.

Johannes Sjöstrand

13. Perturbations of Jordan Blocks

Abstract

In this chapter we shall study the spectrum of a random perturbation of the large Jordan block A ₀, introduced in Sect. 2.4:

$$\displaystyle A_0=\begin {pmatrix}0 &1 &0 &0 &\ldots &0\\ 0 &0 &1 &0 &\ldots &0\\ 0 &0 &0 &1 &\ldots &0\\ . &. &. &. &\ldots &.\\ 0 &0 &0 &0 &\ldots &1\\ 0 &0 &0 &0 &\ldots &0 \end {pmatrix}: {\mathbf {C}}^N\to {\mathbf {C}}^N. $$

Zworski noticed that for every z ∈ D(0, 1), there are associated exponentially accurate quasimodes when N →∞. Hence the open unit disc is a region of spectral instability.
We have spectral stability (a good resolvent estimate) in $\mathbf {C}\setminus \overline {D(0,1)}$, since ∥A ₀∥ = 1.
σ(A ₀) = {0}.

Thus, if A _δ = A ₀ + δQ is a small (random) perturbation of A ₀ we expect the eigenvalues to move inside a small neighborhood of $\overline {D(0,1)}$. In the special case when Qu = (u|e ₁)e _N, where $(e_j)_1^N$ is the canonical basis in C ^N, we have seen in Sect. 2.4 that the eigenvalues of A _δ are of the form

$$\displaystyle \delta ^{1/N}e^{2\pi ik/N},\ k\in \mathbf {Z}/N\mathbf {Z}, $$

so if we fix 0 < δ ≪ 1 and let N →∞, the spectrum “will converge to a uniform distribution on S ¹”.

Johannes Sjöstrand

Spectral Asymptotics for Differential Operators in Higher Dimension

Frontmatter

14. Weyl Asymptotics for the Damped Wave Equation

Abstract

The damped wave equation is closely related to non-self-adjoint perturbations of a self-adjoint operator P of the form

$$\displaystyle P_\epsilon =P+i\epsilon Q. $$

Here, P is a semi-classical pseudodifferential operator of order 0 on L ²(X), where we consider two cases:

X = R ⁿ and P has the symbol P ∼ p(x, ξ) + hp ₁(x, ξ) + ⋯ . in S(m), as in Sect. 6.1, where the description is valid also in the case n > 1. We assume for simplicity that the order function m(x, ξ) tends to + ∞, when (x, ξ) tends to ∞. We also assume that P is formally self-adjoint. Then by elliptic theory (and the ellipticity assumption on P) we know that P is essentially self-adjoint with purely discrete spectrum.
X is a compact smooth manifold with positive smooth volume form dx and P is a formally self-adjoint differential operator, which in local coordinates takes the form,
$$\displaystyle P=\sum _{|\alpha |\le m}a_\alpha (x;h)(hD_x)^\alpha ,\ m>0 $$
where $a_\alpha (x;h)\sim \sum _{k=0}^\infty h^ka_{\alpha ,k}(x)$ in C ^∞ and the “classical” principal symbol
$$\displaystyle p_m(x,\xi )=\sum _{|\alpha |=m}a_{\alpha ,0} (x)\xi ^\alpha , $$
satisfies
$$\displaystyle 0\le p_m(x,\xi )\asymp |\xi |{ }^m , $$
so m has to be even. In this case the semi-classical principal symbol is given by
$$\displaystyle p(x,\xi )=\sum _{|\alpha |\le m}a_{\alpha ,0} (x)\xi ^\alpha . $$

Johannes Sjöstrand

15. Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small Random Perturbations, Results and Outline

Abstract

In this chapter we will state a result asserting that for elliptic semi-classical (pseudo-)differential operators the eigenvalues are distributed according to Weyl’s law “most of the time” in a probabilistic sense. The first three sections are devoted to the formulation of the results and in the last section we give an outline of the proof that will be carried out in Chaps. 16 and 17.

Johannes Sjöstrand

16. Proof I: Upper Bounds

Abstract

In this chapter we study upper bounds on singular values and determinants of certain operators related to P _δ. The bounds are not probabilistic; they only depend on a certain smallness of the perturbation.

Johannes Sjöstrand

17. Proof II: Lower Bounds

Abstract

In this chapter we give a lower bound on $\ln \det S_{\delta ,z}$ which is valid with high probability, and then using also the upper bounds of Chap. 16, we conclude the proof of Theorem 15.3.1 with the help of Theorem 12.1.2.

Johannes Sjöstrand

18. Distribution of Large Eigenvalues for Elliptic Operators

Abstract

In this chapter we consider elliptic differential operators on a compact manifold and rather than taking the semi-classical limit (h →), we let h = 1 and study the distribution of large eigenvalues. Bordeaux Montrieux (Loi de Weyl presque sûre et résolvante pour des opérateurs différentiels non-autoadjoints, thèse, CMLS, Ecole Polytechnique, 2008. https://pastel.archives-ouvertes.fr/pastel-00005367, Ann Henri Poincaré 12:173–204, 2011) studied elliptic systems of differential operators on S ¹ with random perturbations of the coefficients, and under some additional assumptions, he showed that the large eigenvalues obey the Weyl law almost surely. His analysis was based on a reduction to the semi-classical case, where he could use and extend the methods of Hager (Ann Henri Poincaré 7:1035–1064, 2006).

Johannes Sjöstrand

19. Spectral Asymptotics for Symmetric Operators

Abstract

$\mathcal {P}\mathcal {T}$-symmetry has been proposed as an alternative to self-adjointness in quantum physics, see Bender et al. (J Math Phys 40(5):2201–2229, 1999), Bender and Mannheim (Phys Lett A 374(15–16):1616–1620, 2010). Thus for instance, if we consider a Schrödinger operator on R ⁿ,

$$\displaystyle P=-h^2\Delta +V(x), $$

the usual assumption of self-adjointness (implying that the potential V is real valued) can be replaced by that of $\mathcal {P}\mathcal {T}$-symmetry:

$$\displaystyle V\circ \iota =\overline {V}, $$

where ι : R ⁿ →R ⁿ is an isometry with ι ² = 1≠ι. If we introduce the parity operator $\mathcal {P}_\iota u(x)=u(\iota (x))$ and the time reversal operator $\mathcal {T} u=\overline {u}$, then this can be written

$$\displaystyle [P,\mathcal {P}_\iota \mathcal {T} ]=0. $$

Johannes Sjöstrand

20. Numerical Illustrations

Abstract

In this chapter we give some numerical illustrations to the results about the asymptotic distribution of eigenvalues. Such calculations have already been carried out by many people, Trefethen, Trefethen and Embree, Davies, Davies and Hager, Zworski and many others. Numerical calculations with special attention to Weyl asymptotics have been carried out by Hager, Bordeaux Montrieux, Vogel. Many of the illustrations below are therefore well-known and even though we wrote our own Matlab programs, we have clearly benefitted from the preceding works.

Johannes Sjöstrand

Backmatter

Title: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations
Author: Johannes Sjöstrand
Publisher: Springer International Publishing
Electronic ISBN: 978-3-030-10819-9
Print ISBN: 978-3-030-10818-2
DOI: https://doi.org/10.1007/978-3-030-10819-9

Springer Professional

Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

About this book

Table of Contents

Frontmatter

1. Introduction

Basic Notions, Differential Operators in One Dimension

Frontmatter

2. Spectrum and Pseudo-Spectrum

3. Weyl Asymptotics and Random Perturbations in a One-Dimensional Semi-classical Case

4. Quasi-Modes and Spectral Instability in One Dimension

5. Spectral Asymptotics for More General Operators in One Dimension

6. Resolvent Estimates Near the Boundary of the Range of the Symbol

7. The Complex WKB Method

8. Review of Classical Non-self-adjoint Spectral Theory

Some general results

Frontmatter

9. Quasi-Modes in Higher Dimension

10. Resolvent Estimates Near the Boundary of the Range of the Symbol

11. From Resolvent Estimates to Semigroup Bounds

12. Counting Zeros of Holomorphic Functions

13. Perturbations of Jordan Blocks

Spectral Asymptotics for Differential Operators in Higher Dimension

Frontmatter

14. Weyl Asymptotics for the Damped Wave Equation

15. Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small Random Perturbations, Results and Outline

16. Proof I: Upper Bounds

17. Proof II: Lower Bounds

18. Distribution of Large Eigenvalues for Elliptic Operators

19. Spectral Asymptotics for Symmetric Operators

20. Numerical Illustrations

Backmatter

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