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Anish Mallick

Publications and source records attributed to Anish Mallick.

11 recordsLinked to original sources
arXiv2021

Extending Prolog for Quantified Boolean Horn Formulas

Prolog is a well known declarative programming language based on propositional Horn formulas. It is useful in various areas, including artificial intelligence, automated theorem proving, mathematical logic and so on. An active research area for many years is to extend Prolog to larger classes of logic. Some important extensions of it includes the constraint logic programming, and the object oriented logic programming. However, it cannot solve problems having arbitrary quantified Horn formulas. To be precise, the facts, rules and queries in Prolog are not allowed to have arbitrary quantified variables. The paper overcomes this major limitations of Prolog by extending it for the quantified Boolean Horn formulas. We achieved this by extending the SLD-resolution proof system for quantified Boolean Horn formulas, followed by proposing an efficient model for implementation. The paper shows that the proposed implementation also supports the first-order predicate Horn logic with arbitrary quantified variables. The paper also introduces for the first time, a declarative programming for the quantified Boolean Horn formulas.

cs.LO↗
arXiv2020

Global multiplicity bounds and Spectral Statistics Random Operators

In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on $\mathbb{R}$. We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular, we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory. In particular, in higher rank Anderson models with pure-point spectrum, with the randomness having support equal to $\mathbb{R}$, there is a uniform lower bound on spectral multiplicity and in case this is larger than one the local statistics is not Poisson.

math.SP↗
arXiv2020

Regularity of the density of states of Random Schrödinger Operators

In this paper we solve a long standing open problem for Random Schrödinger 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 Hölder continuous $m+1$ st derivative. The required Hölder continuity depends on the fractional moment bounds satisfied by appropriate operator kernels. Our proof of the Random Schrödinger 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 $N^α, α>0$. 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.

math.SP↗
arXiv2018

On multiplicity of spectrum for Anderson type operators with higher rank perturbations

Here, we focus on Anderson type operators over infinite graphs where the randomness acts through higher rank perturbations. We show that for special family of graphs, the operator has non-trivial multiplicity for its pure point spectrum. We, also, show that for some family of graphs, any unitary which fixes the random operator, arising from an automorphism of the graph is identity; but that, for these graphs the spectrum of the random operator has non-trivial multiplicity.

math.SP↗
arXiv2018

Schrödinger operators with decaying randomness - Pure point spectrum

Here we show that for Schrödinger operator with decaying random potential with fat tail single site distribution, the negative spectrum shows a transition from essential spectrum to discrete spectrum. We study the Schrödinger operator $H^ω=-Δ+\displaystyle\sum_{n\in\mathbb{Z}^d}a_nω_nχ_{_{(0,1]^d}}(x-n)$ on $L^2(\mathbb{R}^d)$. Here we take $a_n=O(|n|^{-α})$ for large $n$ where $α>0$, and $\{ω_n\}_{n\in\mathbb{Z}^d}$ are i.i.d real random variables with absolutely continuous distribution $μ$ such that $\frac{dμ}{dx}(x)=O\big(|x|^{-(1+δ)}\big)~as~|x|\to\infty$, for some $δ>0$. We show that $H^ω$ exhibits exponential localization on negative part of spectrum independent of the parameters chosen. For $αδ\leq d$ we show that the spectrum is entire real line almost surely, but for $αδ>d$ we have $σ_{ess}(H^ω)=[0,\infty)$ and negative part of the spectrum is discrete almost surely. In some cases we show the existence of the absolutely continuous spectrum.

math.SP↗
arXiv2018

Spectral statistics of random Schrödinger operator with growing potential

In this work we investigate the spectral statistics of random Schrödinger operators $H^ω=-Δ+\sum_{n\in\mathbb{Z}^d}(1+|n|^α)q_n(ω)|δ_n\rangle\langleδ_n|$, $α>0$ acting on $\ell^2(\mathbb{Z}^d)$ where $\{q_n\}_{n\in\mathbb{Z}^d}$ are i.i.d random variables distributed uniformly on $[0,1]$.

math.SP↗
arXiv2018

Spectral Statistics for one dimensional Anderson model with unbounded but decaying potential

In this work, we study the spectral statistics for Anderson model on $\ell^2(\mathbb{N})$ with decaying randomness whose single site distribution has unbounded support. Here we consider the operator $H^ω$ given by $(H^ωu)_n=u_{n+1}+u_{n-1}+a_nω_n u_n$, $a_n\sim n^{-α}$ and $\{ω_n\}$ are real i.i.d random variables following symmetric distribution $μ$ with fat tail, i.e $μ((-R,R)^c)<\frac{C}{R^δ}$ for $R\gg 1$, for some constant $C$. In case of $α-\frac{1}δ>\frac{1}{2}$, we are able to show that the eigenvalue process in $(-2,2)$ is the clock process.

math.SP↗
arXiv2018

Multiplicity theorem of singular Spectrum for general Anderson type Hamiltonian

In this work, we focus on the multiplicity of singular spectrum for operators of the form $A^ω=A+\sum_{n}ω_n C_n$ 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 $\{ω_n\}_n$ 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 $\sqrt{C_n}(A^ω-z)^{-1}\sqrt{C_n}$ for all $n$ and almost all $(z,ω)$. 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.

math.SP↗
arXiv2016

Multiplicity bound of Singular Spectrum for higher rank Anderson models

In this work, we prove a bound on multiplicity of the singular spectrum for certain class of Anderson Hamiltonians. The class of operator is $H^ω=Δ+\sum_{n\in\mathbb{Z}^d}ω_n P_n$ on the Hilbert space $\ell^2(\mathbb{Z}^d)$, where $Δ$ is discrete laplacian, $P_n$ are projection onto $\ell^2(\{x\in\mathbb{Z}^d:n_il_i<x_i\leq (n_i+1)l_i\})$ for some $l_1,\cdots,l_d\in\mathbb{N}$ and $\{ω_n\}_n$ are i.i.d real bounded random variables following absolutely continuous distribution. We prove that the multiplicity of singular spectrum is bounded above by $2^d-d$ independent of $\{l_i\}_{i=1}^d$. When $l_i+1\not\in 2\mathbb{N}\cup3\mathbb{N}$ for all $i$ and $gcd(l_i+1,l_j+1)=1$ for $i\neq j$, we also prove that the singular spectrum is simple.

math.SP↗
arXiv2015

Jakšić-Last Theorem for Higher Rank Perturbations

We consider the generalized Anderson Model $Δ+\sum_{n\in\mathcal{N}}ω_n P_n$, where $\mathcal{N}$ is a countable set, $\{ω_n\}_{n\in\mathcal{N}}$ are i.i.d random variables and $P_n$ are rank $N<\infty$ projections. For these models we prove theorem analogous to that of Jakšić-Last on the equivalence of the trace measure $σ_n(\cdot)=tr(P_nE_{H^ω}(\cdot)P_n)$ for $n\in\mathcal{N}$ a.e $ω$. Our model covers the dimer and polymer models.

math-ph↗