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Georgi Raikov

Publications and source records attributed to Georgi Raikov.

At least 19 recordsLinked to original sources
arXiv2022

Spectral Asymptotics at Thresholds for a Dirac-type Operator on $\mathbb{Z}^2$

In this article, we provide the spectral analysis of a Dirac-type operator on $\mathbb{Z}^2$ by describing the behavior of the spectral shift function associated with a sign-definite trace-class perturbation by a multiplication operator. We prove that it remains bounded outside a single threshold and obtain its main asymptotic term in the unbounded case. Interestingly, we show that the constant in the main asymptotic term encodes the interaction between a flat band and whole non-constant bands. The strategy used is the reduction of the spectral shift function to the eigenvalue counting function of some compact operator which can be studied as a toroidal pseudo-differential operator.

math.SP↗
arXiv2021

The fate of Landau levels under $δ$-interactions

We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(\mathbb{R}^2)$ whose spectrum consists of infinitely degenerate eigenvalues $Λ_q$, $q \in \mathbb{Z}_+$, and the perturbed operator $H_\upsilon = H_0 + \upsilonδ_Γ$, where $Γ\subset \mathbb{R}^2$ is a regular Jordan $C^{1,1}$-curve, and $\upsilon \in L^p(Γ;\mathbb{R})$, $p>1$, has a constant sign. We investigate ${\rm Ker}(H_\upsilon -Λ_q)$, $q \in \mathbb{Z}_+$, and show that generically $$0 \leq {\rm dim \, Ker}(H_\upsilon -Λ_q) - {\rm dim \, Ker}(T_q(\upsilon δ_Γ)) < \infty,$$ where $T_q(\upsilon δ_Γ) = p_q (\upsilon δ_Γ)p_q$, is an operator of Berezin-Toeplitz type, acting in $p_q L^2(\mathbb{R}^2)$, and $p_q$ is the orthogonal projection on ${\rm Ker}\,(H_0 -Λ_q)$. If $\upsilon \neq 0$ and $q = 0$, we prove that ${\rm Ker}\,(T_0(\upsilon δ_Γ)) = \{0\}$. If $q \geq 1$, and $Γ= \mathcal{C}_r$ is a circle of radius $r$, we show that ${\rm dim \, Ker} (T_q(δ_{\mathcal{C}_r})) \leq q$, and the set of $r \in (0,\infty)$ for which ${\rm dim \, Ker}(T_q(δ_{\mathcal{C}_r})) \geq 1$, is infinite and discrete.

math.SP↗
arXiv2021

Spectral properties of 2D Pauli operators with almost periodic electromagnetic fields

We consider a 2D Pauli operator with almost periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists an almost periodic magnetic potential which generates the magnetic field $b - b_{0}$, $b_{0}$ being the mean value of $b$. Next, we assume that $V = 0$, and investigate the zero modes of $H$. As expected, if $b_{0} \neq 0$, then generically $\operatorname{dim} \operatorname{Ker} H = \infty$. If $b_{0} = 0$, then for each $m \in {\mathbb N} \cup \{ \infty \}$, we construct almost periodic $b$ such that $\operatorname{dim} \operatorname{Ker} H = m$. This construction depends strongly on results concerning the asymptotic behavior of Dirichlet series, also obtained in the present article.

math.SP↗
arXiv2019

Threshold singularities of the spectral shift function for geometric perturbations of magnetic Hamiltonians

We consider the 3D Schrödinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain $Ω_{\rm in} \subset {\mathbb R}^3$. We introduce the Krein spectral shift functions $ξ(E;H_\pm,H_0)$, $E \geq 0$, for the operator pairs $(H_\pm,H_0)$, and study their singularities at the Landau levels $Λ_q : = b(2q+1)$, $q \in {\mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. We show that $ξ(E;H_+,H_0)$ remains bounded as $E \uparrow Λ_q$, $q \in {\mathbb Z}_+$ being fixed, and obtain three asymptotic terms of $ξ(E;H_-,H_0)$ as $E \uparrow Λ_q$, and of $ξ(E;H_\pm,H_0)$ as $E \downarrow Λ_q$. The first two terms are independent of the perturbation while the third one involves the {\em logarithmic capacity} of the projection of $Ω_{\rm in}$ onto the plane perpendicular to $B$.

math.SP↗
arXiv2019

Spectral properties of Landau Hamiltonians with non-local potentials

We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({\mathbb R^2})$, whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues $Λ_q$, $q \in {\mathbb Z}_+$. We perturb $H_0$ by a non-local potential written as a bounded pseudo-differential operator ${\rm Op}^{\rm w}({\mathcal V})$ with real-valued Weyl symbol ${\mathcal V}$, such that ${\rm Op}^{\rm w}({\mathcal V}) H_0^{-1}$ is compact. We study the spectral properties of the perturbed operator $H_{\mathcal V} = H_0 + {\rm Op}^{\rm w}({\mathcal V})$. First, we construct symbols ${\mathcal V}$, possessing a suitable symmetry, such that the operator $H_{\mathcal V}$ admits an explicit eigenbasis in $L^2({\mathbb R^2})$, and calculate the corresponding eigenvalues. Moreover, for ${\mathcal V}$ which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of $H_{\mathcal V}$ adjoining any given $Λ_q$. We find that the effective Hamiltonian in this context is the Toeplitz operator ${\mathcal T}_q({\mathcal V}) = p_q {\rm Op}^{\rm w}({\mathcal V}) p_q$, where $p_q$ is the orthogonal projection onto ${\rm Ker}(H_0 - Λ_q I)$, and investigate its spectral asymptotics.

math-ph↗
arXiv2018

Eigenvalue Asymptotics in a Twisted Waveguide

We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.

math.SP↗
arXiv2018

Lifshits Tails for Squared Potentials

We consider Schrödinger operators with a random potential which is the square of an alloy-type potential. We investigate their integrated density of states and prove Lifshits tails. Our interest in this type of models is triggered by an investigation of randomly twisted waveguides.

math-ph↗
arXiv2018

Lifshits tails for randomly twisted quantum waveguides

We consider the Dirichlet Laplacian $H_γ$ on a 3D twisted waveguide with random Anderson-type twisting $γ$. We introduce the integrated density of states $N_γ$ for the operator $H_γ$, and investigate the Lifshits tails of $N_γ$, i.e. the asymptotic behavior of $N_γ(E)$ as $E \downarrow \inf {\rm supp}\, dN_γ$. In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.

math.SP↗
arXiv2017

Spectral Properties of Harmonic Toeplitz Operators and Applications to the Perturbed Krein Laplacian

We consider harmonic Toeplitz operators $T_V = PV:{\mathcal H}(Ω) \to {\mathcal H}(Ω)$ where $P: L^2(Ω) \to {\mathcal H}(Ω)$ is the orthogonal projection onto ${\mathcal H}(Ω) = \left\{u \in L^2(Ω)\,|\,Δu = 0 \; \mbox{in}\;Ω\right\}$, $Ω\subset {\mathbb R}^d$, $d \geq 2$, is a bounded domain with $\partial Ω\in C^\infty$, and $V: Ω\to {\mathbb C}$ is a suitable multiplier. First, we complement the known criteria which guarantee that $T_V$ is in the $p$th Schatten-von Neumann class $S_p$, by sufficient conditions which imply $T_V \in S_{p, {\rm w}}$, the weak counterpart of $S_p$. Next, we assume that $Ω$ is the unit ball in ${\mathbb R}^d$, and $V = \overline{V}$ is radially symmetric, and investigate the eigenvalue asymptotics of $T_V$ if $V$ has a power-like decay at $\partial Ω$ or $V$ is compactly supported in $Ω$. Further, we consider general $Ω$ and $V \geq 0$ which is regular in $Ω$, and admits a power-like decay of rate $γ> 0$ at $\partial Ω$, and we show that in this case $T_V$ is unitarily equivalent to a pseudo-differential operator of order $-γ$, self-adjoint in $L^2(\partial Ω)$. Using this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for the operator $T_V$. Finally, we introduce the Krein Laplacian $K \geq 0$, self-adjoint in $L^2(Ω)$; it is known that ${\rm Ker}\,K = {\mathcal H}(Ω)$, and the zero eigenvalue of $K$ is isolated. We perturb $K$ by $V \in C(\overlineΩ;{\mathbb R})$, and show that $σ_{\rm ess}(K+V) = V(\partial Ω)$. Assuming that $V \geq 0$ and $V{|\partial Ω} = 0$, we study the asymptotic distribution of the eigenvalues of $K \pm V$ near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator $T_V$.

math.SP↗
arXiv2017

Surface Lifshits tails for random quantum Hamiltonians

We consider Schrödinger operators on $L^{2}({\mathbb R}^{d})\otimes L^{2}({\mathbb R}^{\ell})$ of the form $ H_ω~=~H_{\perp}\otimes I_{\parallel} + I_{\perp} \otimes {H_\parallel} + V_ω$, where $H_{\perp}$ and $H_{\parallel}$ are Schrödinger operators on $L^{2}({\mathbb R}^{d})$ and $L^{2}({\mathbb R}^{\ell})$ respectively, and $ V_ω(x,y)$ : = $\sum_{ξ\in {\mathbb Z}^{d}} λ_ξ(ω) v(x - ξ, y)$, $x \in {\mathbb R}^d$, $y \in {\mathbb R}^\ell$, is a random 'surface potential'. We investigate the behavior of the integrated density of surface states of $H_ω$ near the bottom of the spectrum and near internal band edges. The main result of the current paper is that, under suitable assumptions, the behavior of the integrated density of surface states of $H_ω$ can be read off from the integrated density of states of a reduced Hamiltonian $H_{\perp}+W_ω$ where $W_ω$ is a quantum mechanical average of $V_ω$ with respect to $y \in {\mathbb R}^\ell$. We are particularly interested in cases when $H_{\perp}$ is a magnetic Schrödinger operator, but we also recover some of the results from [24] for non-magnetic $H_{\perp}$.

math-ph↗
arXiv2015

Discrete spectrum of Schrödinger operators with oscillating decaying potentials

We consider the Schrödinger operator $H_{ηW} = -Δ+ ηW$, self-adjoint in $L^2({\mathbb R}^d)$, $d \geq 1$. Here $η$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of $H_{ηW}$ near the origin, and due to the irregular decay of $ηW$, we encounter some non semiclassical phenomena. In particular, $H_{ηW}$ has less eigenvalues than suggested by the semiclassical intuition.

math.SP↗
arXiv2015

Spectral Asymptotics for Waveguides with Perturbed Periodic Twisting

We consider the twisted waveguide $Ω_θ$, i.e. the domain obtained by the rotation of the bounded cross section $ω\subset {\mathbb R}^{2}$ of the straight tube $Ω: = ω\times {\mathbb R}$ at angle $θ$ which depends on the variable along the axis of $Ω$. We study the spectral properties of the Dirichlet Laplacian in $Ω_θ$, unitarily equivalent under the diffeomorphism $Ω_θ\to Ω$ to the operator $H_{θ'}$, self-adjoint in ${\rm L}^2(Ω)$. We assume that $θ' = β- ε$ where $β$ is a $2π$-periodic function, and $ε$ decays at infinity. Then in the spectrum $σ(H_β)$ of the unperturbed operator $H_β$ there is a semi-bounded gap $(-\infty, {\mathcal E}_0^+)$, and, possibly, a number of bounded open gaps $({\mathcal E}_j^-, {\mathcal E}_j^+)$. Since $ε$ decays at infinity, the essential spectra of $H_β$ and $H_{β- ε}$ coincide. We investigate the asymptotic behaviour of the discrete spectrum of $H_{β- ε}$ near an arbitrary fixed spectral edge ${\mathcal E}_j^\pm$. We establish necessary and quite close sufficient conditions which guarantee the finiteness of $σ_{\rm disc}(H_{β-ε})$ in a neighbourhood of ${\mathcal E}_j^\pm$. In the case where the necessary conditions are violated, we obtain the main asymptotic term of the corresponding eigenvalue counting function. The effective Hamiltonian which governs the the asymptotics of $σ_{\rm disc}(H_{β-ε})$ near ${\mathcal E}_j^\pm$ could be represented as a finite orthogonal sum of operators of the form $-μ\frac{d^2}{dx^2} - ηε$, self-adjoint in ${\rm L}^2({\mathbb R})$; here, $μ> 0$ is a constant related to the so-called effective mass, while $η$ is $2π$-periodic function depending on $β$ and $ω$.

math.SP↗
arXiv2013

Scattering in twisted waveguides

We consider a twisted quantum waveguide i.e. a domain of the form Ω_θ : = r_θω\times R, where ω\subset R^2 is a bounded domain, and r_θ= r_θ(x_3) is a rotation by the angle θ(x_3) depending on the longitudinal variable x_3. We investigate the nature of the essential spectrum of the Dirichlet Laplacian H_θ, self-adjoint in L^2 (Ω_θ), and consider related scattering problems. First, we show that if the derivative of the difference θ_1 - θ_2 decays fast enough as |x_3| goes to infinity, then the wave operators for the operator pair (H_{θ_1}, H_{θ_2}) exist and are complete. Further, we concentrate on appropriate perturbations of constant twisting, i.e. θ' = β- ε, with constant β\in R, and εwhich decays fast enough at infinity together with its first derivative. In this case the unperturbed operator corresponding to εis an analytically fibered Hamiltonian with purely absolutely continuous spectrum. Obtaining Mourre estimates with a suitable conjugate operator, we prove, in particular, that the singular continuous spectrum of H_θ, is empty.

math.SP↗
arXiv2013

A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian

We consider the Landau Hamiltonian perturbed by a long-range electric potential $V$. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. First, we obtain an estimate of the rate of the shrinking of these clusters to the Landau levels as the number of the cluster $q$ tends to infinity. Further, we assume that there exists an appropriate $\V$, homogeneous of order $-ρ$ with $ρ\in (0,1)$, such that $V(x) = \V(x) + O(|x|^{-ρ- ε})$, $ε> 0$, as $|x| \to \infty$, and investigate the asymptotic distribution of the eigenvalues within a given cluster, as $q \to \infty$. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the mean-value transform of $\V$.

math.SP↗
arXiv2012

Dirichlet and Neumann Eigenvalues for Half-Plane Magnetic Hamiltonians

Let $H_{0, D}$ (resp., $H_{0,N}$) be the Schroedinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let $H_\ell : = H_{0, \ell} - V$, $\ell =D,N$, where the scalar potential $V$ is non negative, bounded, does not vanish identically, and decays at infinity. We compare the distribution of the eigenvalues of $H_D$ and $H_N$ below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behaviour of the discrete spectrum of $H_\ell$ near $\inf σ_{ess}(H_\ell) = \inf σ(H_{0,\ell})$, $\ell = D,N$. Applying these Hamiltonians, we show that $σ_{disc}(H_D)$ is infinite even if $V$ has a compact support, while $σ_{disc}(H_N)$ could be finite or infinite depending on the decay rate of $V$.

math.SP↗
arXiv2012

Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

We consider the Landau Hamiltonian (i.e. the 2D Schroedinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of the perturbation potential V.

math.SP↗
arXiv2011

Counting function of characteristic values and magnetic resonances

We consider the meromorphic operator-valued function 1-K(z) = 1-A(z)/z where A(z) is holomorphic on the domain D, and has values in the class of compact operators acting in a given Hilbert space. Under the assumption that A(0) is a selfadjoint operator which can be of infinite rank, we study the distribution near the origin of the characteristic values of 1-K(z), i.e. the complex numbers w for which the operator 1-K(w) is not invertible, and we show that generically the characteristic values of 1-K(z) converge to 0 with the same rate as the eigenvalues of A(0). We apply our abstract results to the investigation of the resonances of the operator H = H_0 + V where H_0 is the shifted 3D Schrödinger operator with constant magnetic field of scalar intensity b>0, and V is a real electric potential which admits a suitable decay at infinity. It is well known that the spectrum of H_0 is purely absolutely continuous, coincides with [0,+\infty[, and the so-called Landau levels 2bq with integer q, play the role of thresholds in the spectrum of H_0. We study the asymptotic distribution of the resonances near any given Landau level, and under generic assumptions obtain the main asymptotic term of the corresponding resonance counting function, written explicitly in the terms of appropriate Toeplitz operators.

math.SP↗