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103
Nonselfadjoint harmonic oscillator, compact semigroups and pseudospectra
 J. Operator Theory
"... We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. ..."
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Cited by 10 (2 self)
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We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. The second relies on the fact that the bounded holomorphic semigroup generated by the complex harmonic oscillator is of HilbertSchmidt type in a maximal angular region. In order to show this last property, we deduce a nonselfadjoint version of the classical Mehler’s formula.
Almost Periodic Solutions of First and Second Order Cauchy Problems
 J. Differential Equations
, 1997
"... Almost periodicity of solutions of rst and second order Cauchy problems on the real line is proved under the assumption that the imaginary (resp. real) spectrum of the underlying operator is countable. Related results have been obtained by RuessV~u and Basit. Our proof uses a new idea. It is based ..."
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Cited by 9 (1 self)
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Almost periodicity of solutions of rst and second order Cauchy problems on the real line is proved under the assumption that the imaginary (resp. real) spectrum of the underlying operator is countable. Related results have been obtained by RuessV~u and Basit. Our proof uses a new idea. It is based on a factorisation method which also gives a short proof (of the vectorvalued version) of Loomis' classical theorem, saying that a bounded uniformly continuous function from IR into a Banach space X with countable spectrum is almost periodic if c 0 6 X . Our method can also be used for solutions on the halfline. This is done in a separate paper. 1991 Mathematics subject Classication. 34C28, 44A10, 47D03 Key words and phrases. First and second order Cauchy problems, almost periodic functions, countable spectrum, cosine functions. 0 1 Introduction A central subject in the theory of dierential equations in Banach spaces is to nd criteria for almost periodicity of solutions, see e.g....
A.: Weak Coupling and Continuous Limits for Repeated Quantum Interactions preprint mparc
"... We consider a quantum system in contact with a heat bath consisting in an infinite chain of identical subsystems at thermal equilibrium at inverse temperature β. The time evolution is discrete and such that over each time step of duration τ, the reference system is coupled to one new element of the ..."
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Cited by 8 (2 self)
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We consider a quantum system in contact with a heat bath consisting in an infinite chain of identical subsystems at thermal equilibrium at inverse temperature β. The time evolution is discrete and such that over each time step of duration τ, the reference system is coupled to one new element of the chain only, by means of an interaction of strength λ. We consider three asymptotic regimes of the parameters λ and τ for which the effective evolution of observables on the small system becomes continuous over suitable macroscopic time scales T and whose generator can be computed: the weak coupling limit regime λ → 0, τ = 1, the regime τ → 0, λ 2 τ → 0 and the critical case λ 2 τ = 1, τ → 0. The first two regimes are perturbative in nature and the effective generators they determine is such that a nontrivial invariant subalgebra of observables naturally emerges. The third asymptotic regime goes beyond the perturbative regime and provides an effective dynamics governed by a general Lindblad generator naturally constructed from the interaction Hamiltonian. Conversely, this result shows that one can attach to any Lindblad generator a repeated quantum interactions model whose asymptotic effective evolution is generated by this Lindblad operator. AMS classification numbers: 81Q99
H∞ FUNCTIONAL CALCULUS AND SQUARE FUNCTIONS ON Noncommutative L^Pspaces
, 2006
"... In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semig ..."
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Cited by 8 (3 self)
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In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, qOrnsteinUhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.
Spectral properties of random nonselfadjoint matrices and operators
, 2001
"... We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also desc ..."
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Cited by 7 (4 self)
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We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the nonselfadjoint Anderson model changes suddenly as one passes to the infinite volume limit.
WEYLTITCHMARSH THEORY FOR STURMLIOUVILLE OPERATORS WITH DISTRIBUTIONAL COEFFICIENTS
, 2012
"... We systematically develop Weyl–Titchmarsh theory for singular differential operators on arbitrary intervals (a,b) ⊆ R associated with rather general differential expressions of the type τf = 1 ..."
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Cited by 7 (6 self)
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We systematically develop Weyl–Titchmarsh theory for singular differential operators on arbitrary intervals (a,b) ⊆ R associated with rather general differential expressions of the type τf = 1
Hypoelliptic heat kernel inequalities on Lie groups
, 2005
"... This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associate ..."
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Cited by 6 (1 self)
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This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated “Ricci curvature ” takes on the value − ∞ at points of degeneracy of the semiRiemannian metric associated to the operator. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting. This paper presents recent results for hypoelliptic operators. Malliavin calculus methods transfer the problem to one of determining certain infinite dimensional estimates. Here, the underlying manifold is a Lie group, and the hypoelliptic operators are invariant under left translation. In particular, “L ptype ” gradient estimates hold for p ∈ (1, ∞), and the p = 2 gradient estimate