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72
Continuity properties of Schrödinger semigroups with magnetic fields
 Rev. Math. Phys
"... The objects of the present study are oneparameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Katolike conditions. The configuration space ..."
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Cited by 40 (10 self)
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The objects of the present study are oneparameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Katolike conditions. The configuration space is supposed to be an arbitrary open subset of multidimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show localnormcontinuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownianbridge expectation. Altogether, the article is meant to extend some of the results in B. Simon’s landmark paper (Bull. Amer. Math. Soc. (N.S.) 7, 447–526 (1982)) to nonzero vector potentials and more general configuration
Risk communication
 Proceedings of the national conference on risk communication, Conservation Foundation,Washington, DC
, 1987
"... We consider Schrodinger semigroups e. IH, H =A+V on Iw ” with VcIxl ’ as 1x1rco, OO. We determine the exact power law divergence of I~e‘Hi~p,p and of some IIe‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a for which t ..."
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Cited by 29 (1 self)
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We consider Schrodinger semigroups e. IH, H =A+V on Iw ” with VcIxl ’ as 1x1rco, O<c<[(l/2)(n2)] * with H>O. We determine the exact power law divergence of I~e‘Hi~p,p and of some IIe‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a for which there exists a positive resonance 9 such that Hq = 0, q(x) 1.x‘.:Ta 1991 Academic Press, Inc. 1.
Spherical asymptotics for the rotorrouter model in Z d
 Indiana Univ. Math. J
"... The rotorrouter model is a deterministic analogue of random walk invented by Jim Propp. It can be used to define a deterministic aggregation model analogous to internal diffusion limited aggregation. We prove an isoperimetric inequality for the exit time of simple random walk from a finite region i ..."
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Cited by 25 (11 self)
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The rotorrouter model is a deterministic analogue of random walk invented by Jim Propp. It can be used to define a deterministic aggregation model analogous to internal diffusion limited aggregation. We prove an isoperimetric inequality for the exit time of simple random walk from a finite region in Z d, and use this to prove that the shape of the rotorrouter aggregation model in Z d, suitably rescaled, converges to a Euclidean ball in R d. 1
spectrum, and dynamics for Schrödinger operators on infinite domains
 Duke Math. J
"... 1. Introduction and main results. In this paper we investigate the relations between the rate of decay of solutions of Schrödinger equations, continuity properties of spectral measures of the corresponding operators, and dynamical properties of the corresponding quantum systems. The first main resul ..."
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Cited by 22 (5 self)
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1. Introduction and main results. In this paper we investigate the relations between the rate of decay of solutions of Schrödinger equations, continuity properties of spectral measures of the corresponding operators, and dynamical properties of the corresponding quantum systems. The first main result of this paper shows that, in great generality, certain upper bounds on the rate of growth of L2 norms of generalized
Gaugeability and Conditional Gaugeability
 TRANS. AMER. MATH. SOC
, 2001
"... New Kato classes are introduced for general transient Borel right processes, under which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Greentight measures in the classical Brownian motion case. However the main focus of this paper is on establishing ..."
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Cited by 21 (5 self)
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New Kato classes are introduced for general transient Borel right processes, under which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Greentight measures in the classical Brownian motion case. However the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability. We show that gaugeability, conditional gaugeability and the subcriticality for the associated Schrödinger operators are equivalent for transient Borel right processes with strong duals. for transient Borel standard processes having strong duals. Analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric Markov processes. These analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases. Connections with the positivity of the spectral radii of the associated Schrödinger operators are also established.
INTRINSIC ULTRACONTRACTIVITY OF NONSYMMETRIC DIFFUSION SEMIGROUPS IN BOUNDED DOMAINS
 TOHOKU MATH. J.
, 2008
"... We extend the concept of intrinsic ultracontractivity to nonsymmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of nonsymmetric second order elliptic operators in bounded Lipschitz domains. ..."
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Cited by 21 (18 self)
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We extend the concept of intrinsic ultracontractivity to nonsymmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of nonsymmetric second order elliptic operators in bounded Lipschitz domains.
The integrated density of states for random Schrödinger operators
 in “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday
, 2007
"... Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is di ..."
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Cited by 20 (1 self)
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Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems of current
Spectral Localization by Gaussian Random Potentials in MultiDimensional Continuous Space
, 2000
"... this paper is to contribute to the understanding of spectral localization for random Schrdinger operators in multidimensional Euclidean space ..."
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Cited by 19 (4 self)
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this paper is to contribute to the understanding of spectral localization for random Schrdinger operators in multidimensional Euclidean space