“Nonlinear models in Quantum Mechanics”
Thursday, January 23, 2020, h. 11:30 - Room 2AB45 - Andrea Sacchetti (Università degli Studi di Modena e Reggio Emilia)
In this talk we discuss some recent results for a class of nonlinear models in Quantum Mechanics. In particular we focus our attention to the nonlinear one-dimensional Schrödinger equation (NLS) with a periodic potential and a Stark-type perturbation, in the limit of large periodic potential.
In the first part of the talk we prove the existence of a dense energy spectrum because a cascade of bifurcations of stationary solutions occurs when the ratio between the effective nonlinearity strength and the tilt of the external field increases.
In the second part of the talk we prove the validity of the tight binding approximation, i.e. the reduction of the NLS to a discrete NLS. This model has many interesting features: e.g. the measurement of the value of the gravity acceleration g, using ultracold Strontium atoms confined in a vertical optical lattice.
- Sacchetti A., Nonlinear Stark-Wannier equation, SIAM Journal on Mathematical Analysis (2018).
- Sacchetti A., Derivation of the Tight-Binding Approximation for Time-Dependent Nonlinear Schrödinger Equations, Annales Henri Poincaré (2019).