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.
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).
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.
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.