Accurate Spectral Asymptotics for periodic operators

Victor Ivrii

Displaying similar documents to “Accurate Spectral Asymptotics for periodic operators”

New spectral criteria for almost periodic solutions of evolution equations

Toshiki Naito, Nguyen Van Minh, Jong Son Shin (2001)

Studia Mathematica

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We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form ẋ = A(t)x + f(t) (*), with f having precompact range, which is then applied to find new spectral criteria for the existence of almost periodic solutions with specific spectral properties in the resonant case where $\bar{e^{i s p (f)}}$ may intersect the spectrum of the monodromy operator P of (*) (here sp(f) denotes the Carleman spectrum of f). We show that if (*)...

Spectral projections for the twisted Laplacian

Herbert Koch, Fulvio Ricci (2007)

Studia Mathematica

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Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L = - 1 / 2 \sum_{j = 1}^{n} [(\partial_{x_{j}} + i y_{j}) ² + (\partial_{y_{j}} - i x_{j}) ²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{λ}$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $| | P_{λ} {u | |}_{L^{p} (ℝ^{d})} ≲ λ^{ϱ (p)} {| | u | |}_{L ² (ℝ^{d})}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p...

Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields

Georgi D. Raikov (1999)

Annales de l'institut Fourier

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We consider the Pauli operator $H (μ) : = {(\sum_{j = 1}^{m} σ_{j} (- i \frac{\partial}{\partial x_{j}} - μ A_{j}))}^{2} + V$ selfadjoint in $L^{2} (ℝ^{m}; ℂ^{2})$ , $m = 2, 3$ . Here $σ_{j}$ , $j = 1, ..., m$ , are the Pauli matrices, $A : = (A_{1}, ..., A_{m})$ is the magnetic potential, $μ > 0$ is the coupling constant, and $V$ is the electric potential which decays at infinity. We suppose that the magnetic field generated by $A$ satisfies some regularity conditions; in particular, its norm is lower-bounded by a positive constant, and, in the case $m = 3$ , its direction is constant. We investigate the asymptotic behaviour as $μ \to \infty$ of the number of the eigenvalues of $H (μ)$ smaller...

Spectral Asymptotics for the $\bar{\partial}$ -Neumann Problem

Guy Métivier (1981-1982)

Publications mathématiques et informatique de Rennes

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A.e. convergence of anisotropic partial Fourier integrals on Euclidean spaces and Heisenberg groups

D. Müller, E. Prestini (2010)

Colloquium Mathematicae

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We define partial spectral integrals $S_{R}$ on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by $l o g (2 + L_{α}) f \in L ²$ , where $L_{α}$ is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that $S_{R} f (x)$ converges a.e. to f(x) as R → ∞.

Unique continuation for Schrödinger operators in dimension three or less

Eric T. Sawyer (1984)

Annales de l'institut Fourier

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We show that the differential inequality $| Δ u | \leq v | u |$ has the unique continuation property relative to the Sobolev space $H_{l o c}^{2, 1} (Ω)$ , $Ω \subset R^{n}$ , $n \leq 3$ , if $v$ satisfies the condition $(K_{n}^{loc}) lim_{r \to 0} sup_{x \in K} \int_{| x - y | < r} | x - y |^{2 - n} v (y) d y = 0$ for all compact $K \subset Ω$ , where if $n = 2$ , we replace $| x - y |^{2 - n}$ by $- log | x - y |$ . This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, $H = - Δ + v$ , in the case $n \leq 3$ . The proof uses Carleman’s approach together with the following pointwise inequality valid for all $N = 0, 1, 2, ...$ and any $u \in H_{c}^{2, 1} (R^{3} - {0}),$ $\frac{| u (x) |}{| x |^{N}} \leq C \int_{R^{3}} | x - y |^{- 1} \frac{| Δ u (y) |}{| y |^{N}} d y for a.e. x in R^{3} .$

Weighted norm estimates and $L_{p}$ -spectral independence of linear operators

Peer C. Kunstmann, Hendrik Vogt (2007)

Colloquium Mathematicae

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We investigate the $L_{p}$ -spectrum of linear operators defined consistently on $L_{p} (Ω)$ for p₀ ≤ p ≤ p₁, where (Ω,μ) is an arbitrary σ-finite measure space and 1 ≤ p₀ < p₁ ≤ ∞. We prove p-independence of the $L_{p}$ -spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω,μ); the balls with respect to this semi-metric are required to satisfy a subexponential volume growth condition. We show how previous results on $L_{p}$ -spectral independence...

Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension $2$

Michael Hitrik, Johannes Sjöstrand (2008)

Annales scientifiques de l'École Normale Supérieure

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We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint $h$ -pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable complex window, provided...

On the spectral radius in $L_{1} (G)$

E. Porada (1971)

Colloquium Mathematicae

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On the spectrum of the operator which is a composition of integration and substitution

Ignat Domanov (2008)

Studia Mathematica

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Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator $V_{ϕ} : f (x) \mapsto \int_{0}^{ϕ (x)} f (t) d t$ be defined on L₂[0,1]. We prove that $V_{ϕ}$ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator $V_{ϕ}$ always equals 1.

Some logarithmic function spaces, entropy numbers, applications to spectral theory

Haroske Dorothee

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AbstractIn [18] and [19] we have studied compact embeddings of weighted function spaces on ℝⁿ, $i d : H_{q}^{s} (w (x), ℝ ⁿ) \to L ₚ (ℝ ⁿ)$ , s>0, 1 < q ≤ p< ∞, s-n/q+n/p > 0, with, for example, $w (x) = ⟨ x ⟩^{α}$ , α > 0, or $w (x) = l o g^{β} ⟨ x ⟩$ , β > 0, and $⟨ x ⟩ = {(2 + | x | ²)}^{1 / 2}$ . We have determined the behaviour of their entropy numbers eₖ(id). Now we are interested in the limiting case 1/q = 1/p + s/n. Let $w (x) = l o g^{β} ⟨ x ⟩$ , β > 0. Our results in [18] imply that id cannot be compact for any β > 0, but after replacing the target space Lₚ(ℝⁿ) by a “slightly” larger one, $L ₚ {(l o g L)}_{- a} (ℝ ⁿ)$ , a...

A spectral gap theorem in SU $(d)$

Jean Bourgain, Alex Gamburd (2012)

Journal of the European Mathematical Society

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We establish the spectral gap property for dense subgroups of SU $(d)$ $(d \geq 2)$ , generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU $(2)$ .

On generalized property (v) for bounded linear operators

J. Sanabria, C. Carpintero, E. Rosas, O. García (2012)

Studia Mathematica

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An operator T acting on a Banach space X has property (gw) if $σ_{a} (T) ∖ σ_{S B F ₊ ¯} (T) = E (T)$ , where $σ_{a} (T)$ is the approximate point spectrum of T, $σ_{S B F ₊ ¯} (T)$ is the upper semi-B-Weyl spectrum of T and E(T) is the set of all isolated eigenvalues of T. We introduce and study two new spectral properties (v) and (gv) in connection with Weyl type theorems. Among other results, we show that T satisfies (gv) if and only if T satisfies (gw) and $σ (T) = σ_{a} (T)$ .

Schur Lemma and the Spectral Mapping Formula

Antoni Wawrzyńczyk (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula $σ_{l} (S)$ = ${(λ (s))}_{s \in S} \in ℂ^{S} | {s - λ (s)}_{s \in S}$ generates a proper left ideal $.$ Using the Schur lemma and the Gelfand-Mazur theorem we prove that $σ_{l} (S)$ has the spectral mapping property for sets S of pairwise commuting elements if (i) B is an m-convex algebra with all maximal left ideals closed, or (ii) B is a locally convex Waelbroeck algebra. The right ideal version of this result is also valid.

A direct solver for finite element matrices requiring $O (N log N)$ memory places

Vejchodský, Tomáš

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We present a method that in certain sense stores the inverse of the stiffness matrix in $O (N log N)$ memory places, where $N$ is the number of degrees of freedom and hence the matrix size. The setup of this storage format requires $O (N^{3 / 2})$ arithmetic operations. However, once the setup is done, the multiplication of the inverse matrix and a vector can be performed with $O (N log N)$ operations. This approach applies to the first order finite element discretization of linear elliptic and parabolic problems in triangular...

Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues

Anton Bovier, Véronique Gayrard, Markus Klein (2005)

Journal of the European Mathematical Society

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We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form $- ϵ Δ + \nabla F (\cdot) \nabla$ on $ℝ^{d}$ or subsets of $ℝ^{d}$ , where $F$ is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of...

The density of states of a local almost periodic operator in $ℝ^{ν}$

Andrzej Krupa (2003)

Studia Mathematica

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We prove the existence of the density of states of a local, self-adjoint operator determined by a coercive, almost periodic quadratic form on $H^{m} (ℝ^{ν})$ . The support of the density coincides with the spectrum of the operator in $L ² (ℝ^{ν})$ .

On a binary relation between normal operators

Takateru Okayasu, Jan Stochel, Yasunori Ueda (2011)

Studia Mathematica

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The main goal of this paper is to clarify the antisymmetric nature of a binary relation ≪ which is defined for normal operators A and B by: A ≪ B if there exists an operator T such that $E_{A} (Δ) \leq T * E_{B} (Δ) T$ for all Borel subset Δ of the complex plane ℂ, where $E_{A}$ and $E_{B}$ are spectral measures of A and B, respectively (the operators A and B are allowed to act in different complex Hilbert spaces). It is proved that if A ≪ B and B ≪ A, then A and B are unitarily equivalent, which shows that the relation ≪ is...

Spectral synthesis and operator synthesis

K. Parthasarathy, R. Prakash (2006)

Studia Mathematica

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Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{\infty} (G)$ -submodule X̂ of ℬ(L²(G)) (where $V^{\infty} (G)$ is the weak-* Haagerup tensor product $L^{\infty} (G) \otimes_{w * h} L^{\infty} (G)$ ), define the concept of X̂-operator synthesis and prove that a...

A generalized Kahane-Khinchin inequality

S. Favorov (1998)

Studia Mathematica

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The inequality $ʃ l o g | \sum a_{n} e^{2 π i φ_{n}} | d φ_{1} \dots d φ_{n} \geq C l o g (\sum | a_{n} {|^{2})}^{1 / 2}$ with an absolute constant C, and similar ones, are extended to the case of $a_{n}$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by $e^{2 π i φ}$ .

The third order spectrum of the p-biharmonic operator with weight

Khalil Ben Haddouch, Najib Tsouli, Zakaria El Allali (2014)

Applicationes Mathematicae

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We show that the spectrum of $Δ ²_{p} u + {2 β \cdot \nabla (| Δ u |}^{p - 2} {Δ u) + | β | ² | Δ u |}^{p - 2} Δ u = α m {| u |}^{p - 2} u$ , where $β \in ℝ^{N}$ , under Navier boundary conditions, contains at least one sequence of eigensurfaces.

Strong spectral gaps for compact quotients of products of $PSL (2, ℝ)$ )

Dubi Kelmer, Peter Sarnak (2009)

Journal of the European Mathematical Society

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The existence of a strong spectral gap for quotients $Γ ∖ G$ of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan–Selberg conjectures. If $G$ has no compact factors then for general lattices a spectral gap can still be established, but there is no uniformity and no effective bounds are known. This note is concerned with the spectral...

Well-posedness of second order degenerate differential equations in vector-valued function spaces

Shangquan Bu (2013)

Studia Mathematica

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Using known results on operator-valued Fourier multipliers on vector-valued function spaces, we give necessary or sufficient conditions for the well-posedness of the second order degenerate equations (P₂): d/dt (Mu’)(t) = Au(t) + f(t) (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), (Mu’)(0) = (Mu’)(2π), in Lebesgue-Bochner spaces $L^{p} (, X)$ , periodic Besov spaces $B_{p, q}^{s} (, X)$ and periodic Triebel-Lizorkin spaces $F_{p, q}^{s} (, X)$ , where A and M are closed operators in a Banach space X satisfying D(A) ⊂ D(M)....