Non-normal operators and random perturbations

In modern mathematical physics one often encounters the mathematical theory of partial differential equations which can be used, for example, to model a quantum system such as an electron in an electric or magnetic field. The principal objects describing these systems are called partial differential operators and one of their fundamental quantities is called the spectrum which describes the possible energy levels of the quantum system measured in laboratories. A certain type of these operators, as they may appear for example in the study of the long time behaviour of a quantum particle subject to some dissipative mechanism or in the modelling of fluid dynamics, is highly unstable and sensitive to even tiny perturbations, meaning in particular that the spectrum of these operators can change a lot even when we change the original operator only slightly. Spectral instability is typically an enemy, as for example in numerical analysis when we are interested in finding stable and viable algorithms to numerically compute the spectrum of such an operator since, due to the rounding error, a computer will always look at a small perturbation of the original problem. However, it is also a virtue that will make it possible to prove the results of our project in broad generality. To better understand the structure of the spectra of such operators, it is a natural idea to subject them to tiny random perturbations and study the typical properties of the spectrum, such as the average distribution and the statistical interaction of the spectral points, also called eigenvalues. This means that we study whether in average the eigenvalues tend to repel or attract each other or whether they are placed in an uncorrelated way. With the present project, in collaboration with Maciej Zworski (University of California, Berkeley) and Lazlo Erdös (Institute of Science and Technology Austria), we plan to prove universality of the microscopic statistics of the eigenvalues of such randomly perturbed operators. This means that we first zoom in to the level of the average spacing of the eigenvalues to reveal their microscopic structure and then study their statistical distribution on this level. Our principal aim is to prove that in this case the distribution of the eigenvalues is independent of the details of the actual unperturbed operator and depends only on underlying symmetries. This subject is quite new and some exciting advances have been made recently, however, it has ties to well-studied theories such as the spectral theory of partial differential operators and probability theory. However, to answer the questions of our project, we will need to combine methods from both those subjects and to develop new ones.

Spectral statistics of non-normal operators subject to small random perturbations In modern mathematical physics one often encounters the mathematical theory of partial differential equations which can be used, for example, to model a quantum system such as an electron in an electric or magnetic field. The principal objects describing these systems are called partial differential operators and one of their fundamental quantities is called the spectrum which can describe the possible energy levels of the quantum system measured in laboratories. A certain type of these operators, as they may appear for example in the study of the long time behavior of a quantum particle subject to some dissipative mechanism or in the modeling of fluid dynamics, is highly unstable and sensitive to even small perturbations, meaning in particular that the spectrum of these operators can change a lot even when we change the original operator only slightly. To better understand the structure of the spectra of such operators from a mathematical point of view, it is very natural to subject them to small random perturbations and study the typical properties of their spectra, such as the average distribution and the statistical interaction of the spectral points, also called eigenvalues. This means that we study whether in average the eigenvalues tend to repel or attract each other or whether they are placed in an independent way. The study of the spectra of random operators has a long history which can be traced back to the early 20th century. However, using randomness to study the generic properties of deterministic yet spectrally unstable operators is a quite recent subject with many interesting developments. With this project, hosted by Maciej Zworski (University of California, Berkeley) and by Lazlo Erdös (Institute of Science and Technology Austria), we investigated whether the microscopic properties of the eigenvalues of such randomly perturbed (spectrally unstable) operators are universal. This means that we first zoom in to the level of the average spacing of the eigenvalues to reveal their microscopic structure and then study their statistical distribution on this level. During the course of this project we understood that indeed the microscopic statistical behavior of the eigenvalues exhibits a form of universality. This means that it does in fact not depend anymore on the details of the unperturbed operator but only the underlying fundamental structure of the unperturbed operator and the type of random perturbation, however, not on its probability distribution.