Browsing by Author "Bruneau, Vincent"

Now showing 1 - 5 of 5
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    Counting Function of Magnetic Resonances for Exterior Problems
    (2016) Bruneau, Vincent; Sambou, Diomba
    We study the asymptotic distribution of the resonances near the Landau levels , , of the Dirichlet (resp. Neumann, resp. Robin) realization in the exterior of a compact domain of of the 3D Schrodinger operator with constant magnetic field of scalar intensity . We investigate the corresponding resonance counting function and obtain the main asymptotic term. In particular, we prove the accumulation of resonances at the Landau levels and the existence of resonance-free sectors. In some cases, it provides the discreteness of the set of embedded eigenvalues near the Landau levels.
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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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    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 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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    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.