Classical and Quantum Behavior of the Integrated Density of States for a Randomly Perturbed Lattice

• Published in: Annales Henri Poincaré Vol. 11; no. 6; pp. 1053 - 1083
• Main Authors: Fukushima, Ryoki, Ueki, Naomasa
• Format: Journal Article
• Language: English
• Published: Basel SP Birkhäuser Verlag Basel 01.12.2010; Springer
• Subjects: Classical and Quantum Gravitation; Combinatorics. Ordered structures; Dynamical Systems and Ergodic Theory; Elementary Particles; Exact sciences and technology; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Measure and integration; Operator theory; Order, lattices, ordered algebraic structures; Other topics in mathematical methods in physics; Physics; Physics and Astronomy; Quantum Field Theory; Quantum Physics; Relativity Theory; Sciences and techniques of general use; Theoretical; Tauberian Theorem; Random Potential; Poisson Point Process; Lower Estimate; Integrate Density; Random potential; Density of states; Wiener integral; Asymptotic behavior; Quantum effect; Many-dimensional calculations; Mathematical physics; Lattice
• ISSN: 1424-0637; 1424-0661
• DOI: 10.1007/s00023-010-0051-6

The asymptotic behavior of the integrated density of states for a randomly perturbed lattice at the infimum of the spectrum is investigated. The leading term is determined when the decay of the single site potential is slow. The leading term depends only on the classical effect from the scalar potential. To the contrary, the quantum effect appears when the decay of the single site potential is fast. The corresponding leading term is estimated and the leading order is determined. In the multidimensional cases, the leading order varies in different ways from the known results in the Poisson case. The same problem is considered for the negative potential. These estimates are applied to investigate the long time asymptotics of Wiener integrals associated with the random potentials.