## PhD research topic - Michele Correggi

#### Research areas

Mathematical physics

#### Keywords

Many-body quantum systems, effective theories, superfluidity, quantized vortices, superconductivity, Ginzburg-Landau theory, point interactions, semiclassical limit, anyons.

#### Research topics

All my lines of research are related to mathematical problems emerging from quantum mechanics, in particular in the description of condensed matter systems.

I am thus interested in studying the phenomena of Bose-Einstein condensation and superconductivity, from both the many-body and effective view points, i.e., within standard quantum mechanics or in the framework of the appropriate effective theories (Gross-Pitaevskii theory, Ginzburg-Landau theory, etc.). In the latter case, the main questions can be reformulated as suitable nonlinear minimization problems and the focus is typically put on the structure of any minimizing state (occurrence and distribution of quantized vortices, phase transitions).

The rigorous derivation of the aforementioned effective models, e.g., in a mean-field limit, is also another topic I am presently working on, in both the stationary and dynamic settings. A special case is given by the so called quasi-classical limit, where a quantum system of particles is interacting with a very intense quantized field: the net effect of such an interaction can be described by a suitable effective Schroedinger operator for the particle system.

A peculiar class of Schroedinger operator for quantum systems is given by zero-range interactions, i.e., singular potentials supported on sets of non-zero codimension. The main interest on such models is related to the fact the most of them are explicitly solvable, even in presence of time-dependent interactions. In fact, in the case of fermionic systems, point interactions play also a key role in the so called unitary regime of the Fermi gas.

Finally, the mathematics of identical quantum particles in two dimensions (anyons) is very intriguing because of the complex structure of the configuration space. The analysis of Schroedinger operators associated to anyonic system is another possible line of research.

#### Bibliography

[1] E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, The Mathematics of the Bose Gas and its Condensation, Oberwolfach Seminars 34, Springer, 2005.

[2] M. Correggi, N. Rougerie, Inhomogeneous Vortex Patterns in Rotating Bose-Einstein Condensates, Commun. Math. Phys. 321 (2013), 817–860.

[3] M. Correggi, N. Rougerie, Boundary Behavior of the Ginzburg-Landau Order Parameter in the Surface Superconductivity Regime, Arch. Rational Mech. Anal. 219 (2015), 553–606.

[4] M. Correggi, M. Falconi, Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit, Ann. H. Poincaré published online (2017).

[5] M. Correggi, G. Dell'Antonio, D. Finco, A. Michelangeli, A. Teta, Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions, Rev. Math. Phys. 24 (2012), 1250017.

[6] M. Correggi, D. Lundholm, N. Rougerie, Local Density Approximation for the Almost-bosonic Anyon Gas, Anal. PDE 10 (2017), 1169–1200.