Asymptotics of spectral gaps of the 1D schrodinger operator with Mathieu potential

The one-dimensional Schrödinger operator L(y) = -yn + v(x)y, considered on R with π[pi]-periodic real-valued potential v(x), is self-adjoint, and its spectrum has a gap-band structure- the intervals of continuous spectrum are separated by spectral gaps. In this thesis, we study the asymptotic behaviour of the spectral gaps of L. In the case of the Mathieu potential v(x) = 2a cos (2x), we give an alternative proof of the result of Harrell-Avron-Simon about the precise asymptotics of the lengths of spectral gaps

Asymptotics of spectral gaps of hill and 1D dirac operators

Let L be the Hill operator or the one-dimensional Dirac operator with π-periodic potential considered on the real line R. The spectrum of L has a band-gap structure, that is, the intervals of continuous spectrum alternate with spectral gaps. The endpoints of these gaps are eigenvalues of the same di erential operator L but considered on the interval [0; π] with periodic or antiperiodic boundary conditions. In this thesis considering the Hill and the one-dimensional periodic Dirac operators, we provide precise asymptotics of the spectral gaps in case of speci c potentials that are linear combinations of two exponential terms

Fitted Q-Learning in Mean-field Games

In the literature, existence of equilibria for discrete-time mean field games has been in general established via Kakutani's Fixed Point Theorem. However, this fixed point theorem does not entail any iterative scheme for computing equilibria. In this paper, we first propose a Q-iteration algorithm to compute equilibria for mean-field games with known model using Banach Fixed Point Theorem. Then, we generalize this algorithm to model-free setting using fitted Q-iteration algorithm and establish the probabilistic convergence of the proposed iteration. Then, using the output of this learning algorithm, we construct an approximate Nash equilibrium for finite-agent stochastic game with mean-field interaction between agents.Comment: 22 page

Frequently hypercyclic weighted backward shifts on spaces of real analytic functions

We study frequent hypercyclicity in the case of weighted backward shift operators acting on locally convex spaces of real analytic functions. We obtain certain conditions on frequent hypercyclicity and linear chaoticity of these operators using dynamical transference principles and the frequent hypercyclicity criterion.Publisher versio

Asymptotics of spectral gaps of 1D Dirac operator whose potential is a linear combination of two exponential terms

The one-dimensional Dirac operator L=i((1)(0) (-1)(0))d/dx + ((0)(Q(x)) (0) P-(x)), P,Q is an element of L-2 ([0, pi]), consider on [0, pi] with periodic or antiperiodic boundary conditions, has discrete spectra. For large enough |n|, n is an element of Z, there are two (counted with multiplicity) eigenvalues lambda(-)(n), lambda(+)(n) (periodic if n is even, or antiperiodic if n is odd) such that |lambda(+/-)(n) - n| < 1/2. We study the asymptotics of spectral gaps gamma(n) = lambda(+)(n) - lambda(-)(n) in the case P(x) = ae(-2ix) + Ae(2ix), Q(x) = be(-2ix) + Be-2ix, where a, A, b, B are any complex numbers. We show, for large enough m, that gamma +/- 2m = 0 and gamma 2m+1 = +/- 2 root(Ab)(m)(aB)(m+1)/4(2m)(m!)(2) [1+O(log(2) m/m(2))], gamma-(2m+1) = +/- 2 root(Ab)(m+1)(aB)(m)/4(2m)(m!)(2) [1+O(log(2) m/m(2))]