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L. I. Danilov

Publications and source records attributed to L. I. Danilov.

8 recordsLinked to original sources
arXiv2024

On the spectrum of the Landau Hamiltonian perturbed by a periodic electric potential $V\in H^s_{\mathrm {loc}}({\mathbb R}^2;{\mathbb R})$, $s > 0$

We prove that in a Sobolev space $H^s_{Λ}({\mathbb R}^2;{\mathbb R})$, $s > 0$, of periodic functions with a given period lattice $Λ$, there exists a dense $G_{δ}$-set ${\mathcal O}$ such that the spectrum of the Landau Hamiltonian $H_B + V$ perturbed by any periodic electric potential $V\in {\mathcal O}$ is absolutely continuous for all homogeneous magnetic fields with a rational flux.

math-ph↗
arXiv2009

On absolute continuity of the spectrum of a periodic magnetic Schrödinger operator

We consider the Schrödinger operator in ${\mathbb R}^n$, $n\geq 3$, with the electric potential $V$ and the magnetic potential $A$ being periodic functions (with a common period lattice) and prove absolute continuity of the spectrum of the operator in question under some conditions which, in particular, are satisfied if $V\in L^{n/2}_{\mathrm {loc}}({\mathbb R}^n)$ and $A\in H^q_{\mathrm {loc}}({\mathbb R}^n;{\mathbb R}^n)$, $q>(n-1)/2$.

math-ph↗
arXiv2009

On the spectrum of the periodic Dirac operator

The absolute continuity of the spectrum for the periodic Dirac operator $$ \hat D=\sum_{j=1}^n(-i\frac {\partial}{\partial x_j}-A_j)\hat α_j + \hat V^{(0)}+\hat V^{(1)}, x\in R^n, n\geq 3, $$ is proved given that either $A\in C(R^n;R^n)\cap H^q_{loc}(R^n;R^n)$, 2q > n-2, or the Fourier series of the vector potential $A:R^n\to R^n$ is absolutely convergent. Here, $\hat V^{(s)}=(\hat V^{(s)})^*$ are continuous matrix functions and $\hat V^{(s)}\hat α_j=(-1}^s\hat α_j\hat V^{(s)}$ for all anticommuting Hermitian matrices $\hat α_j$, $\hat α_j^2=\hat I$, s=0,1.

math-ph↗
arXiv2009

On absolute continuity of the spectrum of a d-dimensional periodic magnetic Dirac operator

In this paper, for d > 2, we prove the absolute continuity of the spectrum of a d-dimensional periodic Dirac operator with some discontinuous magnetic and electric potentials. In particular, for d = 3, electric potentials from Zygmund classes $L^3\ln ^{1+δ}L(K)$, $δ>0$, and also ones with Coulomb singularities, with constraints on charges depending on the magnetic potential, are admitted (here K is the fundamental domain of the period lattice).

math-ph↗
arXiv2007

Absence of eigenvalues for the generalized two-dimensional periodic Dirac operator

A generalized two-dimensional periodic Dirac operator is considered, with $L^{\infty}$-matrix-valued coefficients of the first order derivatives and with complex matrix-valued potential. It is proved that if the matrix-valued potential has zero bound relative to the free Dirac operator, then the spectrum of the operator in question contains no eigenvalues.

math-ph↗