By P. Bougerol, Lacroix

Bankruptcy I THE DETERMINISTIC SCHRODINGER OPERATOR 187 1. the variation equation. Hyperbolic constructions 187 2. Self adjointness of H. Spectral houses . a hundred ninety three. Slowly expanding generalized eigenfunctions 195 four. Approximations of the spectral degree 196 two hundred five. The natural element spectrum. A criterion 6. Singularity of the spectrum 202 bankruptcy II ERGODIC SCHRÖDINGER OPERATORS 205 1. Definition and examples 205 2. common spectral houses 206 three. The Lyapunov exponent within the basic ergodie case 209 four. The Lyapunov exponent within the self reliant eas e 211 five. Absence of totally non-stop spectrum 221 224 6. Distribution of states. Thouless formulation 232 7. The natural aspect spectrum. Kotani's criterion eight. Asymptotic homes of the conductance in 234 the disordered cord bankruptcy III THE natural aspect SPECTRUM 237 238 1. The natural aspect spectrum. First facts 240 2. The Laplace remodel on SI(2,JR) 247 three. The natural element spectrum. moment evidence 250 four. The density of states bankruptcy IV SCHRÖDINGER OPERATORS IN A STRIP 2';3 1. The deterministic Schrödinger operator in 253 a strip 259 2. Ergodie Schrödinger operators in a strip three. Lyapunov exponents within the autonomous case. 262 The natural element spectrum (first evidence) 267 four. The Laplace rework on Sp(~,JR) 272 five. The natural aspect spectrum, moment facts vii APPENDIX 275 BIBLIOGRAPHY 277 viii PREFACE This ebook provides elosely similar sequence of leetures. half A, because of P.

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Xk} C GG a Hormander system. Given 0 < a < 0 < 1, 0 < 6 < 1, m E N, I E I(k), IV The Heat Kernel on Nilpotent Lie Groups 46 there exists a constant C such that for every x E G, every R > 0 and every positive solution u of (at + 0)u = 0 in ]0, R[xB(x, R), we have sup XI yEB(x,bV) a ()mu(RY) 7t CR-JIJ/2 yEB(x,6V ) u(R, y). We start with the particular case of stratified groups. , X,} is included in the first slice of some stratification of LG. , k}. 1 yields a constant C independent of v (that is, from x, R and u) such that by vu,(t,1;) = u(Rt, X r CEinf,6)XI (8t)mu(a,t;)

5 If G is not simply connected, there exists a maximal compact subgroup K. K is different from {e}, contained in the centre of G, and G/K is a simply connected nilpotent group. We shall not give the proofs of these facts, but we shall outline some of them. 1 follows thus immediately from the Campbell-Hausdorff formula and from the fact that G is nilpotent. The total degree of the map P is r. 1 that G, endowed with the product P, is a Lie group, and that the exponential map is a homomorphism from (G, P) to G.

Proof We not only have IIStII2. < Ct-n/4(1 + at)n/4eat, `dt>0, but also the same estimate for the adjoint semigroup St*. 2 < + at)n/4eat, Vt > 0. oo < Ct-n/2(1 + at)n/2eat, IISt111 Vt > 0. 9 Remark Suppose that moreover Re (ewBf,f)>-all+lel)IIfII2, 0~~ 0. References and comments The main features of this chapter are due to N. Varopoulos ([132], [136], [139], [140]). He first developed this theory in the setting of Markov chains ([127], [128], [129], [130], [131], [135]), using ideas of Moser ([90], [91]). ~~