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B: Semiclassical states for nonselfadjoint Schrödinger operators
 Commun. Math. Phys
, 1999
"... We prove that the spectrum of certain nonselfadjoint Schrödinger operators is unstable in the semiclassical limit h → 0. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate semiclassical modes of the operator by the JWKB method ..."
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Cited by 32 (10 self)
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We prove that the spectrum of certain nonselfadjoint Schrödinger operators is unstable in the semiclassical limit h → 0. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate semiclassical modes of the operator by the JWKB method for energies far from the spectrum.
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 21 (7 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
Operator Theory and Harmonic Analysis
, 1996
"... Contents 1. Spectral Theory of Bounded Operators (A) Spectra and resolvents of bounded operators on Banach spaces (B) Holomorphic functional calculi of bounded operators 2. Spectral Theory of Unbounded Operators (C) Spectra and resolvents of closed operators in Banach spaces (D) Holomorphic function ..."
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Cited by 16 (6 self)
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Contents 1. Spectral Theory of Bounded Operators (A) Spectra and resolvents of bounded operators on Banach spaces (B) Holomorphic functional calculi of bounded operators 2. Spectral Theory of Unbounded Operators (C) Spectra and resolvents of closed operators in Banach spaces (D) Holomorphic functional calculi of operators of type S!+ 3. Quadratic Estimates (E) Quadratic norms of operators of type S!+ in Hilbert spaces (F) Boundedness of holomorphic functional calculi 4. Operators with Bounded Holomorphic Functional Calculi (G) Accretive operators (H) Operators of type S! and spectral projections 5. Singular Integrals (I) Convolutions and the functional calculus of \Gammai d dx (J) The Hilbert transform and Hardy spaces 6. Calder'onZygmund Theory (K) Maximal functions and the Calder'onZygmund decomposition (L) Singular integral operators 7. Functional Ca
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 12 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
HARDY SPACES ASSOCIATED TO NONNEGATIVE SELFADJOINT OPERATORS SATISFYING DAVIESGAFFNEY ESTIMATES
"... Abstract. Let X be a metric space with doubling measure, and L be a nonnegative, selfadjoint operator satisfying DaviesGaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characte ..."
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Cited by 8 (0 self)
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Abstract. Let X be a metric space with doubling measure, and L be a nonnegative, selfadjoint operator satisfying DaviesGaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characterization, duality of Hardy and BMO spaces. Further specializing to the case that L is a Schrödinger operator on R n with a nonnegative, locally integrable potential, we establish addition characterizations of such Hardy space in terms of maximal functions. Finally, (X) for p> 1, which may or may not coincide with the space L p (X), and show that they interpolate with H 1 L(X) spaces by the complex method. we define Hardy spaces H p L The authors gratefully acknowledge support from NSF as follows: S. Hofmann (DMS
Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials
, 2006
"... We show various L p estimates for Schrödinger operators −∆+V on R n and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen [Sh1]. Our main tools are improved FeffermanPhong inequalities and reverse Hölder estimates for weak solutions of − ∆ + ..."
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Cited by 7 (3 self)
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We show various L p estimates for Schrödinger operators −∆+V on R n and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen [Sh1]. Our main tools are improved FeffermanPhong inequalities and reverse Hölder estimates for weak solutions of − ∆ + V and their gradients.
Plancherel type estimates and sharp spectral multipliers
 J. FUNCT. ANAL
, 2002
"... We study general spectral multiplier theorems for selfadjoint positive definite operators on L²(X, µ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmandertype spectral multiplier theorems follow from the appropriate estimates of the L² norm of the kernel of ..."
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Cited by 7 (0 self)
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We study general spectral multiplier theorems for selfadjoint positive definite operators on L²(X, µ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmandertype spectral multiplier theorems follow from the appropriate estimates of the L² norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hörmandertype spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on R³ and new spectral multiplier theorems for the Laguerre and Hermite expansions.
Heat Kernels and maximal L^p  L^q Estimates for Parabolic Evolution Equations
"... Let A be the generator of an analytic semigroup T on L²(Ω), where Ω is a homogeneous space with doubling property. We prove maximal L p \Gamma L q apriori estimates for the solution of the parabolic evolution equation u 0 (t) = Au(t) + f(t); u(0) = 0 provided T may be represented by a heatk ..."
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Cited by 3 (1 self)
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Let A be the generator of an analytic semigroup T on L²(Ω), where Ω is a homogeneous space with doubling property. We prove maximal L p \Gamma L q apriori estimates for the solution of the parabolic evolution equation u 0 (t) = Au(t) + f(t); u(0) = 0 provided T may be represented by a heatkernel satisfying certain bounds (and in particular a Gaussian bound).
L p Boundedness of Riesz transform related to Schrödinger operators on a manifold
 ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA, CLASSE DI SCIENZE (2009) 725765
, 2009
"... We establish various L p estimates for the Schrödinger operator − ∆ + V on Riemannian manifolds satisfying the doubling property and a Poincaré inequality, where ∆ is the LaplaceBeltrami operator and V belongs to a reverse Hölder class. At the end of this paper we apply our result on Lie groups wit ..."
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Cited by 2 (1 self)
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We establish various L p estimates for the Schrödinger operator − ∆ + V on Riemannian manifolds satisfying the doubling property and a Poincaré inequality, where ∆ is the LaplaceBeltrami operator and V belongs to a reverse Hölder class. At the end of this paper we apply our result on Lie groups with polynomial growth.