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14
Heat kernel estimates for jump processes of mixed types on metric measure spaces
 FIELDS
"... In this paper, we investigate symmetric jumptype processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors dregular sets, which is a class of fractal sets that contains ge ..."
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Cited by 50 (30 self)
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In this paper, we investigate symmetric jumptype processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors dregular sets, which is a class of fractal sets that contains geometrically selfsimilar sets. A typical example of our jumptype processes is the symmetric jump process with jumping intensity e −c0(x,y)x−y � α2 α1 c(α, x, y) ν(dα) x − y  d+α where ν is a probability measure on [α1, α2] ⊂ (0, 2), c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c0(x, y) is a
Perturbation of Dirichlet Forms by Measures
 POTENTIAL ANALYSIS
, 1996
"... Perturbations ofa Dirichlet form 0 by measures/ ~ are studied. The perturbed form 0 # + /z+ is defined for/~ _ in a suitable Kato class and #+ absolutely continuous with respect to capacity. Lpproperties of the corresponding semigroups are derived by approximating # _ by functions. For treating ..."
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Cited by 20 (8 self)
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Perturbations ofa Dirichlet form 0 by measures/ ~ are studied. The perturbed form 0 # + /z+ is defined for/~ _ in a suitable Kato class and #+ absolutely continuous with respect to capacity. Lpproperties of the corresponding semigroups are derived by approximating # _ by functions. For treating #+, a criterion for domination of positive semigroups is proved. If the unperturbed semigroup has Lp Lqsmoothing properties the same is shown to hold for the perturbed semigroup. If the unperturbed semigroup is holomorphic on L ~ the same is shown to be true for the perturbed semigroup, for a large class of measures.
Symmetric jump processes and their heat kernel estimates
 Sci. China Ser. A
"... We survey the recent development of the DeGiorgiNashMoserAronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integrodifferential operators). We focus on the sharp twosided estimates for the transition density functions (or heat kernels) of the proc ..."
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Cited by 9 (6 self)
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We survey the recent development of the DeGiorgiNashMoserAronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integrodifferential operators). We focus on the sharp twosided estimates for the transition density functions (or heat kernels) of the processes, a priori Hölder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integrodifferential operators are mainly probabilistic.
Pointwise Bounds on Eigenfunctions and Wave Packets in NBody Quantum Systems IV
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1978
"... We describe several new techniques for obtaining detailed information on the exponential falloff of discrete eigenfunctions of Nboάy Schrodinger operators. An example of a new result is the bound (conjectured N by Morgan) \ψ(x1...xN)\^Cεxp ( —£α / n) for an eigenfunction ψ of with energy E N. In ..."
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Cited by 8 (3 self)
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We describe several new techniques for obtaining detailed information on the exponential falloff of discrete eigenfunctions of Nboάy Schrodinger operators. An example of a new result is the bound (conjectured N by Morgan) \ψ(x1...xN)\^Cεxp ( —£α / n) for an eigenfunction ψ of with energy E N. In this bound r
Exact Smoothing Properties of Schrödinger Semigroups
, 1997
"... We study Schrodinger semigroups in the scale of Sobolev spaces, and show that, for Kato class potentials, the range of such semigroups in L p has exactly two more derivatives than the potential; this proves a conjecture of B. Simon. We show that eigenfunctions of Schrodinger operators are generica ..."
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Cited by 6 (0 self)
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We study Schrodinger semigroups in the scale of Sobolev spaces, and show that, for Kato class potentials, the range of such semigroups in L p has exactly two more derivatives than the potential; this proves a conjecture of B. Simon. We show that eigenfunctions of Schrodinger operators are generically smoother by exactly two derviatives (in given Sobolev spaces) than their potentials. We give applications to the relation between the potential's smoothness and particle kinetic energy in the context of quantum mechanics, and characterize kinetic energies in Coulomb systems. The techniques of proof invove Leibniz and chain rules for fractional derivatives which are of independent interest, as well as a new characterization of the Kato class. 1 Introduction In this paper we attempt a precise study of the action of Schrodinger semigroups in the scale of Sobolev spaces. Work in this area was begun by B. Simon [Si4] in 1985. He proved a number of positive and negative results on such smooth...
The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 6 (5 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Path Integral Representation for Schrödinger Operators with Bernstein Functions of the Laplacian
"... Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The onetoone correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman ..."
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Cited by 5 (4 self)
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Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The onetoone correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard FeynmanKac formula is taken here by subordinated Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on selfadjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and halfinteger spin values. This approach also allows to propose a notion of generalized Kato class for which hypercontractivity of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived. 1 2 1
Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
, 2009
"... We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples. ..."
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Cited by 4 (4 self)
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We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.