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Dhriti Ranjan Dolai

Publications and source records attributed to Dhriti Ranjan Dolai.

16 recordsLinked to original sources
arXiv2026

Continuity of higher-order derivatives for integrated density of states of the discrete Anderson model with respect to the disorder parameter

We derive quantitative continuity estimates for the higher-order derivatives of the integrated density of states (IDS) with respect to the disorder parameter for the Anderson model on $\ell^2(\mathbb{G})$. Here $\mathbb{G}=\mathbb{Z}^d$ or $\mathbb{B}$, where $\mathbb{B}$ denotes the Bethe lattice. Our results hold in the regime of strong disorder, where entire spectrum is localized. We assume sufficient smoothness of the density of the single site distribution so that the IDS admits higher-order derivatives. More precisely, we establish bounds on the difference between higher-order derivatives of the IDS in terms of the differences in the disorder parameters.

math.SP↗
arXiv2026

CLT for the trace functional of the IDS of magnetic random Schrödinger operators

We consider the existence of the integrated density of states (IDS) of the magnetic Schrödinger operator with a random potential on the Hilbert space \( L^2(\mathbb{R}^d) \), as an analogue of the law of large numbers (LLN) for trace functionals. In this work, we establish an analogue of the central limit theorem (CLT), which describes the fluctuations of the trace functionals of the IDS, for a class of test functions denoted by \( C^1_{d,0}(\mathbb{R}) \). This class consists of real-valued, continuously differentiable functions on \( \mathbb{R} \) that decay at the rate \( O(|x|^{-m}) \) as \( |x| \to \infty \), where \( m > d + 1 \).

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arXiv2024

Central limit theorem for the random variables associated with the IDS of the Anderson model on lattice

We consider the existence of the integrated density of states (IDS) of the Anderson model on the Hilbert space $\ell^2(\mathbb{Z}^d)$ as analogues to the law of large numbers (LLN). In this work, we prove the analogues central limit theorem (CLT) for the collection of random variables associated with the integrated density of states for the class of test functions $C^1_P(\mathbb{R})$, the set of all differentiable (first-order) functions on the real line whose derivative is continuous and has at most polynomial growth. Our work extends the result by Grinshpon-White (J. Spectr. Theory 12 (2022) 591-615), where the CLT is obtained when the test functions are polynomial.

math-ph↗
arXiv2022

Smoothness of integrated density of states and level statistics of the Anderson model when single site distribution is convolution with the Cauchy distribution

In this work we consider the Anderson model on $\ell^2(\mathbb{Z}^d)$ when the single site distribution (SSD) is given by $μ_1 * μ_2$, where $μ_1$ is the Cauchy distribution and $μ_2$ is any probability measure. For this model we prove that the integrated density of states (IDS) is infinitely differentiable irrespective of the disorder strength. Also, we investigate the local eigenvalue statistics of this model in $d\ge 2$, without any assumption on the localization property.

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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

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.

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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↗
arXiv2014

Poisson Statistics for Anderson Model with Singular Randomness

In this work we consider the Anderson model on the $d$-dimensional lattice with the single site potential having singular distribution, mainly $α$-Hölder continuous ones and show that the eigenvalue statistics is Poisson in the region of exponential localization.

math.SP↗
arXiv2014

Spectral statistic for decaying random potentials

We consider Anderson model $H^ω=-Δ+V^ω$ on $\ell^2(\mathbb{Z}^d)$ with decaying random potential. We study the point process $ξ^ω_{L,λ}$ associated with eigenvalues of $H^ω_{Λ_L}$, the retriction of $H^ω$ to the finite cube $Λ_L$. Our result is that the weak limit points of $\{ξ^ω_{L,λ}\}$ are poisson point processes as $L\to\infty$.

math.SP↗
arXiv2013

Level Repulsion for a class of decaying random potentials

In this paper we consider the Anderson model with decaying randomness and show that statistics near the band edges in the absolutely continuous spectrum in dimensions $d \geq 3$ is independent of the randomness and agrees with that of the free part. We also consider the operators at small coupling and identify the length scales at which the statistics agrees with the free one in the limit when the coupling constant goes to zero.

math.SP↗