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.
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.