On the spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb's inequality
- Jon Johnsen
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A study is made of a recent integral identity of B. Helffer and J. Sjöstrand, which for a not yet fully determined class of probability measures yields a formula for the covariance of two functions (of a stochastic variable); in comparison with the Brascamp-Lieb inequality, this formula is a more flexible and in some contexts stronger means for the analysis of correlation asymptotics in statistical mechanics. Using a fine version of the Closed Range Theorem, the identity's validity is shown to be equivalent to some explicitly given spectral properties of Witten-Laplacians on Euclidean space, and the formula is moreover deduced from the obtained abstract expression for the range projection. As a corollary, a generalised version of Brascamp-Lieb's inequality is obtained. For a certain class of measures occuring in statistical mechanics, explicit criteria for the Witten-Laplacians are found from the Persson-Agmon formula, from compactness of embeddings and from the Weyl calculus, which give results for closed range, strict positivity, essential self-adjointness and domain characterisations.
- [Agm78] S. Agmon,Lectures on exponential decay of solution of second order elliptic equations, Math. Notes, vol. 29, Princeton University Press, 1978.
- [BDH89] P. Bolley, M. Dauge, and B. HelfferConditions suffisantes pour l'injection compacte d'espaces de Sobolev à poids, Séminaire équation aux dérivées partielles (France), vol. 1, Université de Nantes, France, 1989, pp. 1–14.
- [BJS98] V. Bach, T. Jecko, and J. Sjöstrand,Correlation asymptotics of classical lattice spin systems with nonconvex Hamilton function at low temperature, 1998, (in preparation, 4 August).
- [BL76] H. J. Brascamp and E. H. Lieb,On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems including inequalities for log concave functions, and with applications to the diffusion equation, J. Funct. Analysis22 (1976), 366–389.
- [BL92] J. Brüning and M. Lesch,Hilbert complexes, J. Funct. Analysis108 (1992), 88–132.
- [dR55] G. de Rham,Variétés différentiables, Hermann, Paris, 1955.
- [He195] Helffer, B.,Spectral properties of the Kac operator in large dimension, Proceedings on mathematical quantum theory II: Schrödinger operators (Providence, R. I.) (Feldman, J. and Froese, R. and Rosen, L. M., ed.), CMR proceedings and lecture notes, vol 8, A. M. S., Providence, R. I., 1995, Proc. of the Canadian Math. Soc. annual sem. on math. quantum theory; Vancouver, August 1993.
- [He197a] B. Helffer,Remarks on the decay of correlations and Witten Laplacians. Analysis of the dependence on the interaction, September 1997, (accepted by Rev. in Math. Physics).
- [He197b] B. Helffer,Remarks on the decay of correlations and Witten Laplacians. Applications to log-Sobolev inequalities, September 1997, (preliminary note).
- [He198] B. Helffer,Remarks on the decay of correlations and Witten Laplacians—the Brascamp—Lieb inequality and semiclassical limit. J. Functional Analysis155 (1998), 571–586.
- [He199] B. Helffer,Spectral theory and applications, lecture notes, January 1999.
- [Hör66] L. Hörmander,Introduction to complex analysis in several variables, 3 ed., North-Holland Mathematical Library, vol. 7 Elsevier, Amsterdam, 1966, (1990).
- [Hör85] L. Hörmander,The analysis of linear partial differential operators Grundlehren der mathematischen Wissenschaften vol. 256, 257, 274, 275, Springer Verlag, Berlin, 1983, (1985).
- [HS94] B. Helffer and J. Sjöstrand,On the correlation for Kac like models in the convex case, J. Stat. Physics74 (1994), 349–409, (already in report no. 9, Mittag Leffler Institute, 1992–93).
- [Jos98] J Jost,Riemannian geometry and geometric analysis, 2. ed. Universitext, Springer, 1998.
- [Kat73] T. Kato,Schrödinger operators with singular potentials, Israel J. Math.13 (1973), 135–148.
- [KM94] J.-M. Kneib and F. Mignot,Equation de Schmoluchowski généralisée, Ann. Math. Pura Appl.(IV)167 (1994), 257–298.
- [LM68] J.-L. Lions and E. Magenes,Problèmes aux limites non homogènes et applications, Editions Dunod, 1968, Engl. translation“Nonhomogeneous boundary problems and applications”, Springer Verlag 1972.
- [NS97] A. Naddaf and T. Spencer,On homogenization and scaling limit of gradient perturbations of a massless free field, Comm. Math. Physics183 (1997), 55–84.
- [Per60] A. Persson,Bounds for the discrete parts of the spectrum of a semi-bounded Schrödinger operator, Math. Scand.8, (1960), 143–153.
- [Sch59] L. Schwartz,Théorie des distributions a valeurs vectorielles, Ann. Institut Fourier7–8 (1957, 1959) 1–141, 1–209.
- [Sch66] L. Schwartz,Théorie des distributions, revised and enlarged ed., Hermann, Paris, 1966.
- [Sim78] C. G. Simader,Essential self-adjointness of Schrödinger operators bounded from below, Math. Z.159 (1978), 47–50.
- [Sjö93] J. Sjöstrand,Potential wells in high dimensions II, more about the one well case, Ann. Inst. Poincaré, Sect. Phys. Th.58 (1993), no. 1, 42–53.
- [Sjö96] J. Sjöstrand,Correlation asymptotics and Witten Laplacians, Algebra and analysis8 (1996), no. 1, 160–191, also in St. Petersburg Math. J.8 (1997), 123–148.
- [Wan99] Wei-Min Wang,Supersymmetry, Witten complex and asymptotics for directional Lyapunov exponents in ℤd, 1999, (preprint), Université de Paris-Sud, Orsay, France).
- [Wit82] E. Witten,Supersymmetry and Morse theory, J. Differential Geometry17 (1982), 661–692.
- On the spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb's inequality
Integral Equations and Operator Theory
Volume 36, Issue 3 , pp 288-324
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- secondary 35Q80
- Jon Johnsen (1)
- Author Affiliations
- 1. Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7E, DK-9220, Aalborg O, Denmark