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The Spectral Gap for the
, 2001
"... We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain with Ising anisotropy ∆. Our main theorem is that there is a nonvanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about ..."
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We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain with Ising anisotropy ∆. Our main theorem is that there is a nonvanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about
GRAPHS WITH SMALL SPECTRAL GAP
, 2013
"... It is conjectured that connected graphs with given number of vertices and minimum spectral gap (i.e., the difference between their two largest eigenvalues) are double kite graphs. The conjecture is confirmed for connected graphs with at most 10 vertices, and, using variable neighbourhood metaheuri ..."
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It is conjectured that connected graphs with given number of vertices and minimum spectral gap (i.e., the difference between their two largest eigenvalues) are double kite graphs. The conjecture is confirmed for connected graphs with at most 10 vertices, and, using variable neighbourhood
Generating spectral gaps by geometry
, 2004
"... Abstract. Motivated by the analysis of Schrödinger operators with periodic potentials we consider the following abstract situation: Let ∆X be the Laplacian on a noncompact Riemannian covering manifold X with a discrete isometric group Γ acting on it such that the quotient X/Γ is a compact manifold. ..."
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Cited by 8 (4 self)
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. We prove the existence of a finite number of spectral gaps for the operator ∆X associated with a suitable class of manifolds X with nonabelian covering transformation groups Γ. This result is based on the nonabelian Floquet theory as well as the MinMaxprinciple. Groups of type I specify a class
On sumsets and spectral gaps
, 2008
"... Suppose that S ⊆ Fp, where p is a prime number. Let λ1,..., λp be the Fourier coefficients of S arranged as follows  ˆ S(0)  = λ1  ≥ λ2  ≥ · · · ≥ λp. Then, as is well known, the smaller λ2  is, relative to λ1, the larger the sumset S + S must be; and, one can work out as a funct ..."
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function of ε and the density θ = S/p, an upper bound for the ratio λ2/λ1  needed in order to guarantee that S + S covers at least (1 − ε)p residue classes modulo p. Put another way, if S has a large spectral gap, then most elements of Fp have the same number of representations as a sum of two
Estimation of spectral gap for elliptic operators
 Trans. AMS
"... Abstract. Let M be a connected, noncompact, complete Riemannian manifold, consider the operator L = ∆+∇V for some V ∈ C 2 (M) with exp[V] integrable w.r.t. the Riemannian volume element. This paper studies the existence of the spectral gap of L. As a consequence of the main result, let ρ be the dist ..."
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Cited by 38 (16 self)
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Abstract. Let M be a connected, noncompact, complete Riemannian manifold, consider the operator L = ∆+∇V for some V ∈ C 2 (M) with exp[V] integrable w.r.t. the Riemannian volume element. This paper studies the existence of the spectral gap of L. As a consequence of the main result, let ρ
Algebraic spectral gaps
, 2011
"... For the onedimensional Schrödinger equation, some real intervals with no eigenvalues (the spectral gaps) may be obtained rather systematically with a method proposed by H. Giacomini and A. Mouchet in 2007. The present article provides some alternative formulation of this method, suggests some possi ..."
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For the onedimensional Schrödinger equation, some real intervals with no eigenvalues (the spectral gaps) may be obtained rather systematically with a method proposed by H. Giacomini and A. Mouchet in 2007. The present article provides some alternative formulation of this method, suggests some
COMPARISON OF METRIC SPECTRAL GAPS
"... Abstract. Let A = (aij) ∈ Mn(R) be an n by n symmetric stochastic matrix. For p ∈ [1,∞) and a metric space (X, dX), let γ(A, dpX) be the infimum over those γ ∈ (0,∞] for which every x1,..., xn ∈ X satisfy 1 n2 n∑ i=1 n∑ j=1 dX(xi, xj) p 6 γ n n∑ i=1 n∑ j=1 aijdX(xi, xj) p. Thus γ(A, dpX) measures t ..."
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Cited by 1 (1 self)
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the magnitude of the nonlinear spectral gap of the matrix A with respect to the kernel dpX: X×X → [0,∞). We study pairs of metric spaces (X, dX) and (Y, dY) for which there exists Ψ: (0,∞) → (0,∞) such that γ(A, dpX) 6 Ψ (γ(A, dpY)) for every symmetric stochastic A ∈Mn(R) with γ(A, dpY) <∞. When Ψ is linear
On the superrigidity of malleable actions with spectral gap
 J. Amer. Math. Soc
"... Abstract. We prove that if a countable group Γ contains a nonamenable subgroup with centralizer infinite and “weakly normal ” in Γ (e.g. if Γ is nonamenable and has infinite center or is a product of infinite groups) then any measure preserving Γaction on a probability space which satisfies certa ..."
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Cited by 79 (7 self)
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certain malleability, spectral gap and weak mixing conditions is cocycle superrigid. We also show that if Γ � X is an arbitrary free ergodic action of such a group Γ and Λ � Y = T Λ is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the associated II1 factors L
Spectral gap for quantum graphs and their connectivity
, 2013
"... The spectral gap for Laplace operators on metric graphs is investigated in relation to graph’s connectivity, in particular what happens if an edge is added to (or deleted from) a graph. It is shown that in contrast to discrete graphs connection between the connectivity and the spectral gap is not on ..."
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Cited by 1 (1 self)
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The spectral gap for Laplace operators on metric graphs is investigated in relation to graph’s connectivity, in particular what happens if an edge is added to (or deleted from) a graph. It is shown that in contrast to discrete graphs connection between the connectivity and the spectral gap
Spectral gap and exponential decay of correlations
 Comm. Math. Phys
"... We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with shortrange interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, ..."
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Cited by 35 (3 self)
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We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with shortrange interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance
Results 1  10
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1,897