We prove that the spectrum of certain non-self-adjoint Schrödinger operators is unstable in the semi-classical limit h → 0. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate semi-classical modes of the operator by the JWKB method for energies far from the spectrum.

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 Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and

2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

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'on--Zygmund Theory (K) Maximal functions and the Calder'on--Zygmund decomposition (L) Singular integral operators 7. Functional Ca

We study general spectral multiplier theorems for self-adjoint positive definite operators on L²(X, µ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmander-type 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 Hörmander-type 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.

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Let A be the generator of an analytic semigroup T on L 2(\Omega\Gamma1 where \Omega is a homogeneous space with doubling property. We prove maximal L p \Gamma L q a-priori 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 heat-kernel satisfying certain bounds (and in particular a Gaussian bound). 1991 Mathematics Subject Classification: 35K22, 58D25, 47D06 1 Introduction Consider an inhomogeneous initial value problem of the form (1:1) u 0 (t) = Au(t) + f(t); t 2 [0; 1) u(0) = u 0 ; where A is the generator of an analytic semigroup on some Banach space X. It is a well known fact that, in general, the derivative u 0 of a solution u of the above Cauchy problem is less regular than the right hand side f , even though the problem is of parabolic type. This phenomena, sometimes called 'loss of regularity', causes problems in the treatment of quasilinear parabolic problems. It is therefore d...

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Abstract. Let X be a metric space with doubling measure, and L be a non-negative, self-adjoint operator satisfying Davies-Gaffney 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 non-negative, 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

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