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Mathematical aspects of physics

Using new techniques of spectral analysis EU-funded mathematicians were able to answer some outstanding questions in the non-self-adjoint extension of singular differential operators.
Mathematical aspects of physics
Until the 1960s, the spectral analysis of non-self-adjoint operators that appear, for example, in phenomenological models of nuclear scattering remained to a large extent terra incognita. The spectrum structure of one-dimensional and three-dimensional Schröedinger operators were known, but there were many difficulties in the study of these operators.

Even a small non-self-adjoint perturbation could produce a point spectrum with extremely rich structure. Standard tools of spectral theory and variational principles were, therefore, not applicable. As a result, although there were many findings from mathematical physics, the deep theoretical understanding of non-self-adjoint operators was missing.

Mathematicians working on the EU-funded project SPECANSO (Spectral analysis of non-selfadjoint and selfadjoint operators – new methods and applications) exploited recent developments in techniques for these differential operators. These included functional models and the use of so-called boundary triples.

Boundary triples allow the introduction of Dirichlet-to-Neumann-type maps to various differential operators. These are entirely determined by boundary data leading to interesting inverse problems. SPECANSO scientists pursued a new approach to the application of functional models based on the boundary triple idea.

The relation between generalised Dirichlet-to-Neumann maps and resolvent sets for which the operators' behaviour can be described by the spectral theorem was also studied. The SPECANSO team's efforts focused on resolvent sets bordered on both sides and considered the operator's adjoint counterpart.

In addition, they considered some first applications of the spectral theorem for self-adjoint operators to dissipative operators, including non-self-adjoint operators. The analysis of ordinary and partial differential equations whose coefficients have local singularities of sufficiently high order led to new phenomena.

Progress made in all the different directions during the SPECANSO project may have implications for neighbouring mathematical branches. In some sense, non-self-adjoint singular operators and particularly, the Schrödinger operators have always served as a testing ground for methods and theories.

Related information


Spectral analysis, differential operators, nuclear scattering, Schrödinger operators, mathematical physics
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