Hery Randriamaro (Madagaskar): Spectral Theory

Probevorlesung im Rahmen des Habilitationsverfahrens

Abstract:

According to the literature, the oldest problem in spectral theory was dealt with in the context of sound waves around 550 BC when Pythagoras discovered the relationship between the length of a vibrating string and the musical pitch it produces. Mersenne refined this discovery in 1636 by noting that the vibrating string produces, in addition to its fundamental tone, a simultaneous set of overtones. A century after, d’Alembert explained the overtone phenomenon by developing the wave equation as a model for the motion of string. This is the first historical case of an observable spectral phenomenon being explained in terms of differential equations.

In parallel, the concept of the spectrum of a matrix appeared in the late eighteenth century when Lagrange defined the moments of inertia of a rigid body in terms of the characteristic values of a matrix.

The connection between the spectral aspect of matrices and of differential operators became explicit only in the early 1900s when Hilbert introduced the name spectral theory while using inner product spaces to study integral operators. His research laid the foundation for the modern development of spectral theory.

Our lecture precisely exposes some basic concepts of that development. We describe the Hilbert and Sobolev spaces which are respectively the spaces on which the operators are defined and the function spaces which allow to analyze operators. Afterwards, we talk about lineaments of operators such as compactness and self-adjointness. Then, we state fundamental results on spectral theory like the spectral theorem. Spectral theory has many applications, however we only focus on the Laplacian with boundary conditions and the Schrödinger operators.