Browsing by Author "Raikov, Georgi"

Now showing 1 - 16 of 16
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    Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian
    (2013) Pushnitski, Alexander; Raikov, Georgi; Villegas-Blas, Carlos
    We consider the Landau Hamiltonian (i.e. the 2D Schrodinger 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.
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    Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials
    (EUROPEAN MATHEMATICAL SOC, 2011) Bruneau, Vincent; Miranda, Pablo; Raikov, Georgi
    We consider the unperturbed operator H-0 = (-i del - A)(2) + W, self-adjoint in L-2(R-2). Here A is a magnetic potential which generates a constant magnetic field b > 0, and the edge potential W is a non-decreasing non-constant bounded function depending only on the first coordinate x is an element of R of (x, y) is an element of R-2. Then the spectrum of H-0 has a band structure and is absolutely continuous; moreover, the assumption lim(x ->infinity)(W(x) - W(-x)) < 2b implies the existence of infinitely many spectral gaps for H-0. We consider the perturbed operators H-+/- = H-0 +/- V where the electric potential V is an element of L-infinity(R-2) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H-+/- in the spectral gaps of H-0. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian involves a pseudo-differential operator with generalized anti-Wick symbol equal to V. Further, we restrict our attention on perturbations V of compact support and constant sign. We establish a geometric condition on the support of V which guarantees the finiteness of the number of the eigenvalues of H-+/- in any spectral gap of H-0. In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of H+ (resp. H-) to the lower (resp. upper) edge of a given spectral gap, is Gaussian.
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    Discrete spectrum of quantum Hall effect Hamiltonians II: Periodic edge potentials
    (IOS PRESS, 2012) Miranda, Pablo; Raikov, Georgi
    We consider the unperturbed operator H-0 := (-i del-A)(2) + W, self-adjoint in L-2(R-2). Here A is a magnetic potential which generates a constant magnetic field b > 0, and the edge potential W = (W) over bar is a T-periodic non-constant bounded function depending only on the first coordinate x is an element of R of (x, y) is an element of R-2. Then the spectrum sigma(H-0) of H-0 has a band structure, the band functions are bT-periodic, and generically there are infinitely many open gaps in sigma(H-0). We establish explicit sufficient conditions which guarantee that a given band of sigma(H-0) has a positive length, and all the extremal points of the corresponding band function are non-degenerate. Under these assumptions we consider the perturbed operators H-+/- = H-0 +/- V where the electric potential V is an element of L-infinity(R-2) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H-+/- in the spectral gaps of H-0. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian could be interpreted as a 1D Schrodinger operator with infinite-matrix-valued potential. Further, we restrict our attention on perturbations V of compact support. We find that there are infinitely many discrete eigenvalues in any open gap in the spectrum sigma(H-0), and the convergence of these eigenvalues to the corresponding spectral edge is asymptotically Gaussian.
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    Eigenvalue Asymptotics in a Twisted Waveguide
    (TAYLOR & FRANCIS INC, 2009) Briet, Philippe; Kovarik, Hynek; Raikov, Georgi; Soccorsi, Eric
    We consider a twisted quantum wave guide i.e., a domain of the form :=r x where 2 is a bounded domain, and r=r(x3) is a rotation by the angle (x3) depending on the longitudinal variable x3. We are interested in the spectral analysis of the Dirichlet Laplacian H acting in L2(). We suppose that the derivative [image omitted] of the rotation angle can be written as [image omitted](x3)=-epsilon(x3) with a positive constant and epsilon(x3) L|x3|-, |x3|. We show that if L0 and (0,2), or if LL00 and =2, 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.
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    Lifshits Tails for Randomly Twisted Quantum Waveguides
    (2018) Kirsch, Werner; Krejcirik, David; Raikov, Georgi
    We consider the Dirichlet Laplacian on a 3D twisted waveguide with random Anderson-type twisting . We introduce the integrated density of states for the operator , and investigate the Lifshits tails of , i.e. the asymptotic behavior of as . In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.
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    Quantization of Edge Currents along Magnetic Barriers and Magnetic Guides
    (2011) Dombrowski, Nicolas; Germinet, Francois; Raikov, Georgi
    We investigate the edge conductance of particles submitted to an Iwatsuka magnetic field, playing the role of a purely magnetic barrier. We also consider magnetic guides generated by generalized Iwatsuka potentials. In both cases, we prove quantization of the edge conductance. Next, we consider magnetic perturbations of such magnetic barriers or guides and prove stability of the quantized value of the edge conductance. Further, we establish a sum rule for edge conductances. Regularization within the context of disordered systems is discussed as well.
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    Resonances and spectral shift function near the Landau levels
    (ANNALES DE L INSTITUT FOURIER, 2007) Bony, Jean Francois; Bruneau, Vincent; Raikov, Georgi
    We consider the 3D Schrodinger operator H = H-0 + V where H-0 = (-i del - A)(2) - b, A is a magnetic potential generating a constant magneticfield of strength b > 0, and V is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of H admits a meromorphic extension from the upper half plane to an appropriate Riemann surface M, and define the resonances of H as the poles of this meromorphic extension. We study their distribution near any fixed Landau level 2bq, q is an element of N. First, we obtain a sharp upper bound of the number of resonances in a vicinity of 2bq. Moreover, under appropriate hypotheses, we establish corresponding lower bounds which imply the existence of an infinite number of resonances, or the absence of resonances in certain sectors adjoining 2bq. Finally, we deduce a representation of the derivative of the spectral shift function (SSF) for the operator pair (H, H-0) as a sum of a harmonic measure related to the resonances, and the imaginary part of a holomorphic function. This representation justifies the Breit-Wigner approximation, implies a trace formula, and provides information on the singularities of the SSF at the Landau levels.
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    Spectral asymptotics at thresholds for a Dirac-type operator on Z2
    (2023) Miranda, Pablo; Parra, Daniel; Raikov, Georgi
    In this article, we provide the spectral analysis of a Dirac-type operator on 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. (c) 2022 Elsevier Inc. All rights reserved.
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    Spectral Properties of 2D Pauli Operators with Almost-Periodic Electromagnetic Fields
    (2019) Bony, Jean-Francois; Espinoza, Nicolas; Raikov, Georgi
    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 a magnetic potential which generates the magnetic field b - b(0), where b(0) is the mean value of b. Next, we assume that V = 0, and investigate the zero modes of H. As expected, if b(0 )not equal 0, then generically dim Ker H = infinity. If b(0) = 0, then for each m is an element of N boolean OR {infinity}, we construct an almost-periodic b such that dim Ker H = m. This construction depends strongly on results concerning the asymptotic behavior of Dirichlet series, also obtained in the present article.
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    Spectral properties of a magnetic quantum Hamiltonian on a strip
    (IOS PRESS, 2008) Briet, Philippe; Raikov, Georgi; Soccorsi, Eric
    We consider a 2D Schrodinger operator H(0) with constant magnetic field, on a strip of finite width. The spectrum of H(0) is absolutely continuous, and contains a discrete set of thresholds. We perturb H(0) by an electric potential V which decays in a suitable sense at infinity, and study the spectral properties of the perturbed operator H = H(0) + V. First, we establish a Mourre estimate, and as a corollary prove that the singular continuous spectrum of H is empty, and any compact subset of the complement of the threshold set may contain at most a finite set of eigenvalues of H, each of them having a finite multiplicity. Next, we introduce the Krein spectral shift function (SSF) for the operator pair (H, H(0)). We show that this SSF is bounded on any compact subset of the complement of the threshold set, and is continuous away from the threshold set and the eigenvalues of H. The main results of the article concern the asymptotic behaviour of the SSF at the thresholds, which is described in terms of the SSF for a pair of effective Hamiltonians.
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    Spectral properties of harmonic Toeplitz operators and applications to the perturbed Krein Laplacian
    (2018) Bruneau, Vincent; Raikov, Georgi
    We consider harmonic Toeplitz operators T-V = PV :H(Omega) -> H(Omega) where P : L-2(Omega) -> H(Omega) is the orthogonal projection onto H(Omega) = {u is an element of L-2 (Omega)) vertical bar Delta u = 0 in Omega}, Omega subset of R-d, d >= 2, is a bounded domain with boundary partial derivative Omega is an element of C-infinity and V : Omega -> C is an appropriate multiplier. First, we complement the known criteria which guarantee that T-V is in the pth Schatten-von Neumann class S-p, by simple sufficient conditions which imply T-V is an element of S-p(,w), the weak counterpart of S-p. Next, we consider symbols V >= 0 which have a regular power-like decay of rate & nbsp;gamma > 0 at partial derivative Omega, and we show that T-V is unitarily equivalent to a classical pseudo-differential operator of order-gamma, self-adjoint in L-2 (partial derivative Omega). Utilizing this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for T-V, and establish a sharp remainder estimate. Further, we assume that Omega is the unit ball in R-d, and V = (V) over bar is compactly supported in Omega, and investigate the eigenvalue asymptotics of the Toeplitz operator T-V. Finally, we introduce the Krein Laplacian K, self-adjoint in L-2 (Omega), perturb it by a multiplier V is an element of C((Omega) over bar; R), and show that sigma(ess)(K + V) = V (partial derivative Omega). Assuming that V >= 0 and V-vertical bar partial derivative Omega = 0, we study the asymptotic distribution of the discrete spectrum of K +/- V near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator T-V.
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    Spectral properties of Landau Hamiltonians with non-local potentials
    (2020) Cardenas, Esteban; Raikov, Georgi; Tejeda, Ignacio
    We consider the Landau Hamiltonian H-0, self-adjoint in L-2 (R-2), whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues Lambda(q), q is an element of Z(+). We perturb H-0 by a non-local potential written as a bounded pseudo-differential operator Op(w)(V) with real-valued Weyl symbol V, such that Op(w)(V)H-0(-1) is compact. We study the spectral properties of the perturbed operator H-V = H-0 Op(w)(V). First, we construct symbols V, possessing a suitable symmetry, such that the operator H-V admits an explicit eigenbasis in L-2 (R-2), and calculate the corresponding eigenvalues. Moreover, for V which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of H-V adjoining any given Lambda(q). We find that the effective Hamiltonian in this context is the Toeplitz operator T-q(V) = p(q)Op(w)(V)p(q), where p(q) is the orthogonal projection onto Ker(H-0 - Lambda I-q), and investigate its spectral asymptotics.
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    Splitting of the Landau levels by magnetic perturbations and Anderson transition in 2D-random magnetic media
    (2010) Dombrowski, Nicolas; Germinet, Francois; Raikov, Georgi
    In this paper we consider a Landau Hamiltonian perturbed by a random magnetic potential of Anderson type. For a given number of bands, we prove the existence of both strongly localized states at the edges of the spectrum and dynamical delocalization near the center of the band in the sense that wave packets travel at least at a given minimum speed. We provide explicit examples of magnetic perturbations that split the Landau levels into full intervals of spectrum.
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    The fate of Landau levels under δ-interactions
    (2022) Behrndt, Jussi; Holzmann, Markus; Lotoreichik, Vladimir; Raikov, Georgi
    We consider the self-adjoint Landau Hamiltonian H-0 in L-2(R-2) whose spectrum consists of infinitely degenerate eigenvalues Lambda(q), q is an element of Z(+), and the perturbed Landau Hamiltonian H-upsilon = H-0 + upsilon delta(Gamma), where Gamma subset of R-2 is a regular Jordan C-1,C-1-curve and upsilon is an element of L-p(Gamma; R), p > 1, has a constant sign. We investigate ker(H-upsilon - Lambda(q)), q is an element of Z(+), and show that generically
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    The Fate of the Landau Levels under Perturbations of Constant Sign
    (OXFORD UNIV PRESS, 2009) Klopp, Frederic; Raikov, Georgi
    We show that the Landau levels cease to be eigenvalues if we perturb the 2D Schrodinger operator with a constant magnetic field, by bounded electric potentials of fixed sign. We also show that, if the perturbation is not of fixed sign, then any Landau level may be an eigenvalue of the perturbed problem.
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    Threshold Singularities of the Spectral Shift Function for Geometric Perturbations of Magnetic Hamiltonians
    (2020) Bruneau, Vincent; Raikov, Georgi
    We consider the Schrodinger operator H0with constant magnetic field B of scalar intensity b>0self-adjoint in L2(R3) and its perturbations H+ (resp., H-obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain omega in subset of R3 We introduce the Krein spectral shift functions xi(Ex37e;H +/-,H0) for the operator pairs (H +/-,H0)and study their singularities at the Landau levels ?q:=b(2q+1)which play the role of thresholds in the spectrum of H0 We show that xi(Ex37e;H+,H0)remains bounded as E up arrow?qbeing fixed, and obtain three asymptotic terms of xi(Ex37e;H-,H0) as E up arrow?q$$E \uparrow \Lambda _q$$\end{document}, and of xi(Ex37e;H +/-,H0)as E down arrow?qThe first two divergent terms are independent of the perturbation, while the third one involves the logarithmic capacity of the projection of omega inonto the plane perpendicular to B.