# Seminars & Events for Mathematical Physics Seminar

##### Extended States in a Lifshitz Tail Regime for Random Operators on Trees

We will discuss the spectral properties of random operators on regular tree graphs. The models have have been among the earliest studied for Anderson localization, and they continue to attract attention because of analogies with localization issues for many particles. The talk will focus on the location of the mobility edge. Somewhat surprisingly, a resonance mechanism will be proven to cause the appearance of absolutely continuous spectrum in a regime extending well beyond the energy band of the operator's non-random hopping term. For weak disorder, this includes a Lifshitz tail regime of very low density of states.

##### Trace Formulas for Large Random d-Regular Graphs

Trace formulas for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulas depend on a parameter (w) which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w = 1, the only periodic orbits which contribute are the non back-scattering orbits, and the smooth part in the trace formula coincides with the Kesten-McKay expression. As (w) deviates from unity, non vanishing weights are assigned to the periodic walks with back-scatter, and the smooth part is modified in a consistent way. The trace formulas presented in this talk can be used as tools for showing the connection between the spectral properties of d-regular graphs and the theory of random matrices.

##### Geometric methods for nonlinear quantum many-body systems

Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body SchrÃ¶dinger operators. In this talk I will present a formalism which also allows to study nonlinear systems. I will in particular define a weak topology on many-body states, which appropriately describes the physical behavior of the system in the case of lack of compactness, that is when some particles are lost at infinity. As an application I prove the existence of multi-polaron systems in the Pekar- Tomasevich approximation, in a certain regime for the coupling constant.

##### Geometry of quantum response in open systems

I shall describe a theory of adiabatic response for controlled open systems governed by Lindblad evolutions. The theory gives quantum response a geometric interpretation induced from the geometry of Hilbert space projections. For a two level system the metric turns out to be the Fubini-Study metric and the symplectic form the adiabatic curvature. Nice things happen when the metric and symplectic structures are compatible so the space of controls is Kahler. I shall give examples of compatible physical systems. Work based on joint work with Fraas, Graf and Kenneth.