Results 1  10
of
61
Geometric Ergodicity and Hybrid Markov Chains
, 1997
"... Various notions of geometric ergodicity for Markov chains on general state spaces exist. In this paper, we review certain relations and implications among them. We then apply these results to a collection of chains commonly used in Markov chain Monte Carlo simulation algorithms, the socalled hybrid ..."
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Cited by 83 (26 self)
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Various notions of geometric ergodicity for Markov chains on general state spaces exist. In this paper, we review certain relations and implications among them. We then apply these results to a collection of chains commonly used in Markov chain Monte Carlo simulation algorithms, the socalled hybrid chains. We prove that under certain conditions, a hybrid chain will "inherit" the geometric ergodicity of its constituent parts. 1 Introduction A question of increasing importance in the Markov chain Monte Carlo literature (Gelfand and Smith, 1990; Smith and Roberts, 1993) is the issue of geometric ergodicity of Markov chains (Tierney, 1994, Section 3.2; Meyn and Tweedie, 1993, Chapters 15 and 16; Roberts and Tweedie, 1996). However, there are a number of different notions of the phrase "geometrically ergodic", depending on perspective (total variation distance vs. in L 2 ; with reference to a particular V function; etc.). One goal of this paper is to review and clarify the relationship...
A Minimax Principle For The Eigenvalues In Spectral Gaps
 J. London Math. Soc
, 1997
"... . A minimax principle is derived for the eigenvalues in the spectral gap of a possibly nonsemibounded self adjoint operator. It allows us to bound the nth eigenvalue of the Dirac operator with Coulomb potential from below by the nth eigenvalue of a semibounded Hamiltonian which is of interest in ..."
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Cited by 35 (4 self)
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. A minimax principle is derived for the eigenvalues in the spectral gap of a possibly nonsemibounded self adjoint operator. It allows us to bound the nth eigenvalue of the Dirac operator with Coulomb potential from below by the nth eigenvalue of a semibounded Hamiltonian which is of interest in the context of stability of matter. As a second application we show that the Dirac operator with suitable nonpositive potential has at least as many discrete eigenvalues as the Schrodinger operator with the same potential. 1. Introduction The minimax principle provides a variational characterization of all eigenvalues below (or above) the essential spectrum of a a self adjoint operator that is bounded below (above) (see, e.g., the books by Courant and Hilbert [2], Chapter VI, x1.4, or by Reed and Simon [14], chapter XIII.1 ). It allows one to estimate, say, the nth eigenvalue without a priori knowledge on the spectrum or eigenfunctions, which might be the main reason why it is one of the...
Oneparticle (improper) states in Nelson's massless model
, 2002
"... In the oneparticle sector of Nelson's massless model, the oneparticle states are constructed for an arbitrarily small infrared cuto in the interaction term of the Hamiltonian of the system. The performed method is a constructive one which exploits only regular perturbation theory, by a suitable i ..."
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Cited by 19 (2 self)
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In the oneparticle sector of Nelson's massless model, the oneparticle states are constructed for an arbitrarily small infrared cuto in the interaction term of the Hamiltonian of the system. The performed method is a constructive one which exploits only regular perturbation theory, by a suitable iteration scheme. The disappearance of oneparticle states is showed in the limit of no infrared regularization. Constructive features, as regularity in some parameters, are also inquired.
Rates of convergence for everywherepositive Markov chains
, 1995
"... It is often useful to know that the distribution of a Markov process converges to a stationary distribution, and if possible to know how rapidly convergence takes place. Such rates of convergence are of particular interest when running stochastic algorithms such ..."
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Cited by 17 (13 self)
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It is often useful to know that the distribution of a Markov process converges to a stationary distribution, and if possible to know how rapidly convergence takes place. Such rates of convergence are of particular interest when running stochastic algorithms such
Selfadjointness and spectral properties of a pseudorelativistic Hamiltonian due to Brown and Ravenhall
 School of Mathematics, University of Wales
, 1997
"... Some properties of a pseudorelativistic Hamiltonian describing a one electron atom  an appropriately projected Dirac operator with Coulomb potential  proposed by Brown and Ravenhall are given. Selfadjointness is investigated and the explicit form of the Friedrichs extension is given. The behavio ..."
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Cited by 12 (0 self)
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Some properties of a pseudorelativistic Hamiltonian describing a one electron atom  an appropriately projected Dirac operator with Coulomb potential  proposed by Brown and Ravenhall are given. Selfadjointness is investigated and the explicit form of the Friedrichs extension is given. The behavior near the critical coupling constant fl c is described and the essential spectrum is determined in the case of fl = fl c . 1 Introduction To include relativistic effects in the description of a one electron atom with coupling constant fl ? 0, Brown and Ravenhall proposed the operator B := + ` D \Gamma fl jxj ' + (1) selfadjointly realized in the Hilbert space H = + \Gamma L 2 (R 3 )\Omega C 4 \Delta . Here, D := \Gammaiff \Delta r + fim is the Dirac operator and + := (0;1) (D) the projection on the positive spectral subspace of D. The 4 \Theta 4matrices ff j , j 2 f1; 2; 3g, and fi are the four Dirac matrices in standard representation, fl = ffZ with the nuclear char...
Binding conditions for atomic Nelectron systems in nonrelativistic QED
 Ann. Henri Poincaré
"... Abstract. We examine the binding conditions for atoms in nonrelativistic QED, and prove that removing one electron from an atom requires a positive energy. As an application, we establish the existence of a ground state for the Helium atom. Dedicated to Professor G. Zhislin, on the occasion of his ..."
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Cited by 12 (5 self)
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Abstract. We examine the binding conditions for atoms in nonrelativistic QED, and prove that removing one electron from an atom requires a positive energy. As an application, we establish the existence of a ground state for the Helium atom. Dedicated to Professor G. Zhislin, on the occasion of his seventieth birthday. 1.
Nonvariational approximation of discrete eigenvalues of selfadjoint operators
 IMA J. Numer. Anal
"... Abstract. We establish sufficiency conditions in order to achieve approximation to discrete eigenvalues of selfadjoint operators in the secondorder projection method suggested recently by Levitin and Shargorodsky, [15]. We find explicit estimates for the eigenvalue error and study in detail two co ..."
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Cited by 11 (5 self)
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Abstract. We establish sufficiency conditions in order to achieve approximation to discrete eigenvalues of selfadjoint operators in the secondorder projection method suggested recently by Levitin and Shargorodsky, [15]. We find explicit estimates for the eigenvalue error and study in detail two concrete model examples. Our results show that, unlike the majority of the standard methods, secondorder projection strategies combine nonpollution and approximation at a very high level of generality. 1.
Asymptotic Variance and Convergence Rates of NearlyPeriodic MCMC Algorithms
, 2001
"... We consider nearlyperiodic chains, which may have excellent functionalestimation properties but poor distributional convergence rate. We show how simple modications of the chain (involving using a random number of iterations) can greatly improve the distributional convergence of the chain. We prov ..."
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Cited by 11 (4 self)
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We consider nearlyperiodic chains, which may have excellent functionalestimation properties but poor distributional convergence rate. We show how simple modications of the chain (involving using a random number of iterations) can greatly improve the distributional convergence of the chain. We prove various theoretical results about convergence rates of the modied chains. We also consider a number of examples. 1.
Nonperturbative Mass and Charge Renormalization in Relativistic Nophoton QED
 Commun. Math. Phys
"... Abstract. Starting from a formal Hamiltonian as found in the physics literature – omitting photons – we define a renormalized Hamiltonian through charge and mass renormalization. We show that the restriction to the oneelectron subspace is welldefined. Our construction is nonperturbative and does n ..."
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Cited by 7 (5 self)
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Abstract. Starting from a formal Hamiltonian as found in the physics literature – omitting photons – we define a renormalized Hamiltonian through charge and mass renormalization. We show that the restriction to the oneelectron subspace is welldefined. Our construction is nonperturbative and does not use a cutoff. The Hamiltonian is relevant for the description of the Lamb shift in muonic atoms. 1.
DECAY PROPERTIES OF SPECTRAL PROJECTORS WITH APPLICATIONS TO ELECTRONIC STRUCTURE
, 2010
"... Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the ..."
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Cited by 6 (1 self)
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Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential offdiagonal decay (‘nearsightedness’) for the density matrix of gapped systems at zero electronic temperature in both orthogonal and nonorthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for nonmetallic systems. Our theory also allows us to treat the case of density matrices for arbitrary systems at finite electronic temperature, including metals. Other possible applications are also discussed.