Schrödinger operators and spectral theory

An eigenstate of the hydrogen atom. With 45% probability, the solid body contains the electron

The study of the relationship between geometry and spectral properties is one of the oldest subjects of human interest. More than two thousand years ago the Pythagoreans had already discovered connections between the length of a string and the tone it produced. Nowadays, one meets situations in most areas of Physics and Engineering, where it is important to ascertain how shape determines the frequencies of vibrations and their distribution.

Many problems in geometry, number theory and dynamical systems could be interpreted in terms of spectral properties of Partial Differential Operators (PDO). For example, the classical problem concerning the number of integer points in a circle is equivalent to the asymptotical behaviour of the number of eigenvalues of a Dirichlet Laplacian. It is well known that closed geodesics on a Riemannian manifold are closely related to the discrete spectrum of the corresponding Laplace-Beltrami operator. The respective approximative eigenfunctions are usually called quasi-modes. Chaotic behaviour of Hamiltonian trajectories correspond to a non-clustering behaviour of the eigenvalues.

The spectral analysis of differential operators plays a crucial role in the area of Mathematical Physics and, in  particular, in Quantum Mechanics. Many phenomena can be described in terms of the discrete and continuous spectrum of a linear operator. When studying the discrete spectrum one is often interested in regimes where a certain parameter is either very large or very small. A mathematically rigorous analysis usually requires not only the study of the asymptotic behaviour of the relevant quantities, but also a careful bound on approximation errors. This is where spectral estimates play a central part in the proof. The celebrated inequalities by Lieb-Thirring, for example, are of importance in the theory of the Stability of Matter and in the Turbulence Theory.