On the shape of the ground state eigenvalue density of a random hill's equation

Communications on Pure and Applied Mathematics

Volumn 59, Issue 7, 2006, Pages 935-976

  Cambronero, Santiago ^a,c   Rider, Brian ^b,c   Ramírez, José ^a  

^a UNIVERSIDAD DE COSTA RICA   (Costa Rica)

^b DUKE UNIVERSITY   (United States)

^c UNIVERSITY OF COLORADO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33744546387     PISSN: 00103640     EISSN: 00103640     Source Type: Journal    
DOI: 10.1002/cpa.20104     Document Type: Article

References (24)

1. Adler, R. An introduction to continuity, extrema, and related topics for general Gaussian processes. Institute of Mathematical Statistics Lecture Notes-Monograph Series, 12. Institute of Mathematical Statistics, Hay ward, Calif., 1990
2. Ben Arous, G.; Deuschel, J.-D.; Stroock, D. W. Precise asymptotics in large deviations. Bull. Sci. Math 117 (1993), no. 1, 107-124.
3. Bolthausen, E. Laplace approximations for sums of independent random vectors. II. Degenerate maxima and manifolds of maxima. Probab. Theory Related Fields 76 (1987), no. 2, 67-206.
4. Byrd, P.; Friedman, M. D. Handbook of elliptic integrals for engineers and physicists. Springer, Berlin-Göttingen-Heidelberg, 1954.
5. Cambronero, S.; McKean, H. P. The ground state eigenvalue of Hill's equation with white noise potential. Comm. Pure Appl. Math. 52 (1999), no. 10, 1277-1294.
6. Ellis, R. S.; Rosen, J. S. Asymptotic analysis of Gaussian integrals, II: manifolds of minimum points. Comm. Math Phys. 82 (1981/82), no. 2, 153-181.
7. Frisch, H. L.; Lloyd, S. P. Electron levels in a one-dimensional lattice. Phys. Rev. 120 (1960), no. 4, 1175-1189.
8. Fukushima, M.; Nakao, S. On the spectra of the Schrödinger operator with a white Gaussian noise potential. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (1976/77), no. 3, 267-274.
9. Halperin, B. I. Green's functions for a particle in a one-dimensional random potential. Phys. Rev. (2) 139 (1965), A104-A117.
10. Hochstadt, H. Asymptotic estimates for the Sturm-Liouville spectrum. Comm. Pure. Appl. Math 14 (1961), 749-764.
11. Hochstadt, H. Functiontheoretic properties of the discriminant of Hill's equation. Math. Z. 82 (1963), 237-242.
12. Ince, E. L. The periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60 (1940), 47-63.
13. Kac, M. On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc. 49 (1943), 314-320.
14. Kusuoka, S.; Stroock, D. W. Precise asymptotics of certain Wiener functionals. J. Fund. Anal. 99 (1991), no. 1, 1-74.
15. Kusuoko, S.; Stroock, D. W. Asymptotics of certain Wiener functionals with degenerate extrema. Comm. Pure. Appl. Math. 47 (1994), no. 4, 477-501.
16. Lifshits, I. M.; Gredeskul, S. A.; Pastur, L. A. Introduction to the theory of disordered systems. Wiley, New York, 1988.
17. Magnus, W.; Winkler, S. Hill's equation. Interscience Tracts in Pure and Applied Mathematics, No. 20. Wiley, New York-London-Sydney, 1996.
18. McKean, H. P. A limit law for the ground state of Hill's equation. J. Stat. Phys. 74 (1994), no. 5-6, 1227-1232.
19. McKean, H. P.; van Moerbeke, P. The spectrum of Hill's equation. Invent. Math. 30 (1975), no. 3, 217-274.
20. Merkl, F. Quenched asymptotics of the ground state energy of random Schrödinger operators with scaled Gibbsian potentials. Probab. Theory Relat. Fields 126 (2003), no. 3, 307-338.
21. Rice, S. O. Mathematical analysis of random noise. Bell Sys. Tech. J. 23 (1944), 282-332.
22. Shepp, L. A. Radon-Nikodym derivatives of Gaussian measures. Ann. Math. Statist. 37 (1966), 321-354.
23. Sznitman, A.-S. Brownian motion, obstacles and random media. Springer Monographs in Mathematics. Springer, Berlin-Heidelberg, 1998.
24. Varadhan, S. R. S. Lectures on diffusion problems and partial differential equations. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 64. Tata Institute of Fundamental Research, Bombay, 1980. 0977-1041