Mathematical Physics/PDE Seminar

In Spring 2021, the Math.Physics/PDE seminar meets Tuesdays 5:00 -6:15 pm online on Zoom. For more information please contact Boris Vainberg

 

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April 20, 2021      

        S. Molchanov will continue his talk on the sparse potentials.

 

April 13, 2021      

       S. Molchanov will give a talk “Introduction to the spectral theory of lattice Schrodinger operators with sparse potentials    

 

Abstarct. The talk will present several results. In particular, we will describe the essential spectrum of lattice Schrodinger operators with sparse potentials and the point component of the spectral measure of the Schrodinger operator outside of the spectrum of the Laplacian.  We will also discuss the open problems and possible approaches to their solutions.

 

April 6, 2021

        S. Molchanov will continue to talk on “Introduction to the spectral theory of Schrodinger operators with sparse potentials”    

 

Abstract: Kapitsa pendulum. Example of a band-gap spectrum. Review of results on the 1-D theory (Pearson, Kiselev-Last Simon, Molchanov, Cook-Holt-Molchanov).

 

 March 30, 2021

        S. Molchanov will give a talk “Introduction to the spectral theory of Schrodinger operators with sparse potentials”    

                                         

Abstract: In the classical spectral theory of the Schrodinger operators H=-Δ+V(x) (i.e. in the quantum mechanics) there are two fundamental models: 

a)     V(x) vanishes at infinity (may be, with additional restriction on the rate of the decay of the potential). Examples. Hydrogen atom H, here V(x)=-c/|x|, general atoms and molecules, scattering theory.

b)    Periodic potentials. Examples. Theory of the ideal crystals, their heat and electric conductivities, theory of metals and semiconductors.

 

Periodic potentials are also related to the problem of stability of the mechanical system (Kapitsa pendulum). What happens between these two classical cases? The simplest intermediate models are the operators with the sparse potentials where there are infinitely many elementary scatterers (bumps) such that the distances between the bumps are growing to infinity. These models demonstrate many new effects.

 

March 23, 2021

     O. Safronov will continue to talk on “Discrete spectrum of a periodic Schrodinger operator perturbed by a decaying impurity potential”

 

Abstract:  First, we explain how compact operators naturally appear in problems where one studies the flow of eigenvalues through a  fixed  point  λ. In particular, if  H(t)=H-t V where H is a differential operator and V is a positive decaying  function, then one can reduce the study of eigenvalues of H(t)  to the study of the operator W(H- λ )^{-1}W where W^2=V. The latter operator turns out to be compact. After that we will discuss applications of this reduction principle to the case of a periodic operator H.

 

March 16, 2021

     O. Safronov will give a talk “Discrete spectrum of a periodic Schrodinger operator perturbed by a decaying impurity potential”

 

Abstract.  Let  H  be a periodic  Schrodinger operator and let  V be a positive fast decaying  function  on R^d.   We consider the family of operators H(t)= H-tV. We study the number N(t) of eigenvalues of H(t) in a fixed interval [a,b] consisting of regular  points of H. We obtain an asymptotic formula for N(t) as t goes to infinity. However, the limit in this asymptotic formula is understood in some integral Cesaro-like sense.