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An Introduction To The Theory Of Point Processes

Functions of One Complex Variable

We will introduce now the notions of correlation functions and their Fourier transforms which play an important role in various applications of the theory of dynamical systems. Suppose {T t } is a flow on a measure space (M,M,μ) and f ∈ L 2 (M,M,μ). By the correlation function corresponding to f we mean the function b f (t) = ∫ M f(T t x)f(x)dμ A number of statistical properties of the dynamical system may be characterized by the limit behavior of the differences [b f (t) — (∫ M f dμ)2]- In the case of mixing, these expressions tend to zero as t → ∞. If the convergence is fast enough one may write the above expressions in the form 

$${b_f}\left( t \right) - {\left( {\int_M {fd\mu } } \right)^2} = \int_{ - \infty }^\infty {\exp \left( {i\lambda t} \right) \cdot {\rho _f}\left( \lambda \right)d\lambda } $$

. The function ρ f (λ) in this representation is called the spectral density of. The set of those λ for which ρ f (λ) is essentially non-zero for typical ∫, characterizes in some sense the frequencies playing the crucial role in the dynamics of the system under consideration. It is sometimes said that the system “produces a noise” on this set. The investigation of the behavior of the functions ρ f (t), as well as of their analogs ρ f (n) n ∈ Z, in the case of discrete time, is very important both for theory and for applications.

Spectral Theory of Dynamical Systems

Anderson model with decaying randomness: mobility edge: mobility edge

OF DISSERTATION Eigenvalue Statistics and Localization for Random Band Matrices with Fixed Width and Wegner Orbital Model Abstract: We discuss two models from the study of disordered quantum systems. The first is the Random Band Matrix with a fixed band width and Gaussian or more general disorder. The second is the Wegner n-orbital model. We establish that the point process constructed from the eigenvalues of finite size matrices converge to a Poisson Point Process in the limit as the matrix size goes to infinity. The proof is based on the method of Minami for the Anderson tight-binding model. As a first step, we expand upon the localization results by Schenker and Peled-Schenker-Shamis-Sodin to account for complex energies. We use the fractional moment method of Aizenman-Molchanov to derive these bounds. In addition, we establish convergence and smoothness of the density of states functions by modifying estimates of Dolai-Krishna-Mallick to allow for unbounded random variables. From there we follow the Daley and Vere-Jones criteria for establishing the convergence of the eigenvalue point process to the Poisson Point process. The analysis is first presented for the band matrix with adjustments for the orbital model following after. Other properties of these models such as ergodicity and Lyapunov exponents are discussed.

Eigenvalue Statistics and Localization for Random Band Matrices with Fixed Width and Wegner Orbital Model

In this paper we solve a long standing open problem for Random Schrodinger operators on 
$$L^2({\mathbb {R}}^d)$$

 with i.i.d single site random potentials. We allow a large class of free operators, including magnetic potential, however our method of proof works only for the case when the random potentials satisfy a complete covering condition. We require that the supports of the random potentials cover 
$${\mathbb {R}}^d$$

 and the bump functions that appear in the random potentials form a partition of unity. For such models, we show that the Density of States (DOS) is m times differentiable in the part of the spectrum where exponential localization is valid, if the single site distribution has compact support and has Holder continuous 
$$m+1$$

 st derivative. The required Holder continuity depends on the fractional moment bounds satisfied by appropriate operator kernels. Our proof of the Random Schrodinger operator case is an extensions of our proof for Anderson type models on 
$$\ell ^2(\mathbb {G})$$

, 
$$\mathbb {G}$$

 a countable set, with the property that the cardinality of the set of points at distance N from any fixed point grows at some rate in 
$$N^\alpha , \alpha >0$$

. This condition rules out the Bethe lattice, where our method of proof works but the degree of smoothness also depends on the localization length, a result we do not present here. Even for these models the random potentials need to satisfy a complete covering condition. The Anderson model on the lattice for which regularity results were known earlier also satisfies the complete covering condition.

Regularity of the Density of States of Random Schrödinger Operators

In this work, we focus on the multiplicity of singular spectrum for operators of the form $A^\omega=A+\sum_{n}\omega_n C_n$ on a separable Hilbert space $\mathcal{H}$, for a self-adjoint operator $A$ and a countable collection $\{C_n\}_{n}$ of non-negative finite rank operators. When $\{\omega_n\}_n$ are independent real random variables with absolutely continuous distributions, we show that the multiplicity of singular spectrum is almost surely bounded above by the maximum algebraic multiplicity of eigenvalues of $\sqrt{C_n}(A^\omega-z)^{-1}\sqrt{C_n}$ for all $n$ and almost all $(z,\omega)$. The result is optimal in the sense that there are operators where the bound is achieved. Using this, we also provide effective bounds on multiplicity of singular spectrum for some special cases.

Multiplicity theorem of singular Spectrum for general Anderson type Hamiltonian

We consider random Schrodinger operators on $\ell ^{2}(\mathbb {Z}^{d})$ with α-Holder continuous (0<α≤1) single site distribution. In localized regime, we study the distribution of eigenfunctions in space and energy simultaneously. In a certain scaling limit, we prove limit points are Poisson.

Eigenfunction statistics for Anderson model with Hölder continuous single site potential

In this work we investigate the spectral statistics of Random Schr\"odinger operators $H^\omega=-\Delta+\sum_{n\in\mathbb{Z}^d}(1+|n|^\alpha)q_n(\omega)|\delta_n\rangle\langle\delta_n|$, acting on $\ell^2(\mathbb{Z}^d)$ and $\{q_n\}_{n\in\mathbb{Z}^d}$ are i.i.d random variables distributed uniformly on $[0,1]$.

Spectral Statistics of Random Schrödinger Operators with Unbounded Potentials

In this work we investigate the spectral statistics of random Schr\"{o}dinger operators $H^\omega=-\Delta+\sum_{n\in\mathbb{Z}^d}(1+|n|^\alpha)q_n(\omega)|\delta_n\rangle\langle\delta_n|$, $\alpha>0$ acting on $\ell^2(\mathbb{Z}^d)$ where $\{q_n\}_{n\in\mathbb{Z}^d}$ are i.i.d random variables distributed uniformly on $[0,1]$.

Anish Mallick

Papers

Regularity of the Density of States of Random Schrödinger Operators

Multiplicity theorem of singular Spectrum for general Anderson type Hamiltonian

Eigenfunction statistics for Anderson model with Hölder continuous single site potential

Spectral Statistics of Random Schrödinger Operators with Unbounded Potentials

Spectral statistics of random Schr\"{o}dinger operator with growing potential