Shmuel Agmon

Lectures on elliptic boundary value problems

This book, which is a new edition of a book originally published in 1965, presents an introduction to the theory of higher-order elliptic boundary value problems. The book contains a detailed study of basic problems of the theory, such as the problem of existence and regularity of solutions of higher-order elliptic boundary value problems. It also contains a study of spectral properties of operators associated with elliptic boundary value problems. Weyls law on the asymptotic distribution of eigenvalues is studied in great generality.

Asymptotic formulas with remainder estimates for eigenvalues of elliptic operators

We propose to discuss in this lecture a number of results related to the problem of eigenvalue distribution of elliptic operators. We start with some classical results. Let Δ be the Laplacian in Rn and consider the eigenvalue problem:

Perturbation of embedded eigenvalues in the generalized

\begin{array}{*{20}c}{ - \Delta = \lambda {\text{u}}} & {{\text{in}}\,\,\,\Omega \,,}\\{{\text{u}} = 0} & {{\text{on}}\,\,\partial \Omega \,,}\\\end{array}

-body problem

(1)

On Positive Solutions of Elliptic Equations with Periodic Coefficients in N, Spectral Results and Extensions to Elliptic Operators on Riemannian Manifolds

Asymptotic formulas with remainder estimates are derived for spectral functions of general elliptic operators. The estimates are based on asymptotic expansion of resolvent kernels in the complex plane.

On perturbation of embedded eigenvalues

We discuss the perturbation of continuum eigenvalues without analyticity assumptions. Among our results, we show that generally a small perturbation removes these eigenvalues in accordance with Fermis Golden Rule. Thus, generically (in a Baire category sense), the Schrödinger operator has no embedded non-threshold eigenvalues.

Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I

Publisher Summary This chapter discusses the properties of positive solutions and related spectral results for elliptic operators with periodic coefficients on ℝ n . It also discusses the extensions of the results to a larger class of elliptic operators on certain noncompact Riemannian manifolds. The chapter focuses on periodic case. Whenever the elliptic equation Pu = 0 admits a positive solution on ℝ n , it also admits a positive exponential type solution. The chapter describes the family of positive solutions of Pu = 0 that are exponentials. It presents the various properties of this family. Some results in spectral theory of elliptic operators with periodic coefficients that are based on the knowledge of the family of positive exponential solutions are discussed in the chapter. The chapter also describes the extensions of the results to a more general class of elliptic operators defined on certain noncompact Riemannian manifolds. There are close connections between the spectral properties of second order elliptic operators and the properties of positive solutions of elliptic equations.

On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems

It is well known that in a proper setup eigenvalues which belong to the discrete spectrum are stable. This property is the basis of the perturbation theory for such eigenvalues. On the other hand, the behavior of eigenvalues which are embedded in the continuous spectrum is completely different. Such eigenvalues may be very unstable under perturbations. A striking example of instability of embedded eigenvalues was given by Colin de Verdiere [4]. Consider the Laplace operator Δ g on a non-compact hyperbolic surface M having a finite area, g the metric. It is well known that the self-adjoint realization of Δ g (denoted by the same symbol) has a continuous spectrum which is the half-line: (−∞, − 1/4]. The discrete spectrum of Δg is a finite set. However, there are many interesting examples where Δ g has infinitely many eigenvalues embedded in the continuous spectrum. That these examples are exceptional in some sense is shown by the following theorem due to Colin de Verdiere.