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PHYSICAL REVIEW E 68, 065205 R

2003

Low-rank perturbations and the spectral statistics of pseudointegrable billiards
Thomas Gorin*
¨ Theoretische Quantendynamik, Physikalisches Institut, Universitat Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany

Jan Wiersig
¨ ¨ Institut fur Theoretische Physik, Universitat Bremen, Otto-Hahn-Allee, D-28359 Bremen,Germany Received 18 March 2003; published 24 December 2003 ¨ We present an efficient method to solve Schrodinger’s equation for perturbations of low rank. The method is ideally suited for systems with short range interactions or quantum billiards. It involves a secular equation of low dimension, which directly returns the level counting function. For illustration, we calculate the number variancefor two pseudointegrable quantum billiards: the barrier billiard and a right triangle billiard. In this way, we obtain precise estimates for the level compressibility in the semiclassical high energy limit. In both cases, our results confirm recent theoretical predictions, based on periodic orbit summation, disregarding diffractive orbits. DOI: 10.1103/PhysRevE.68.065205 PACS number s : 05.45.Mt,02.70. c, 03.65.Ge

Consider a bound quantum system with Hamiltonian H H 0 W, where the eigenbasis of H 0 is known, and the perturbation W is non-negative nonpositive and of low rank. Then, our method allows to obtain the level counting function, by solving an eigenvalue problem of the dimension which is equal to the rank of the perturbation. This possibility is an important further developmentof the first implementation in Ref. 1 . The class of systems which fulfill the above requirements is large. Quite evidently, few particle systems with short range residual interactions, e.g., Ref. 2 , fall into this category. Quantum billiards can also be considered. To see this, choose for H 0 an integrable billiard B0 , which encloses the billiard B of interest. Then, the boundary can be modeled bya potential with a one-dimensional -shaped profile, which is of low rank in the Hilbert space of H 0 . As a result, the spectrum of H contains the desired eigenvalues of B and those of its complement B0 B. They can be separated with the help of an appropriate observable, and in some cases, e.g., the Sinai billiard, and the examples considered below, the separation is given beforehand. Weillustrate our method with two examples, the barrier billiard 3 , and the right triangle billiard 4 with acute angle /5. Both examples are two-dimensional, pseudointegrable polygon billiards 4,5 . While pseudointegrable systems have enough constants of motion to assure local integrability, singularities in the Hamiltonian flow allow invariant surfaces with genus larger than one. This introduces some kind ofrandomness into the classical dynamics, though the Lyapunov exponent is equal to zero, everywhere. The nonstandard topology of the invariant surfaces leads to an algebraic dispersion of nearby trajectories 6 . In the spirit of quantum-classical correspondence, there have been numerous efforts to study the implications of pseudointegrability on the quantum spectrum 1,5,7–10 . In Refs. 8,11 it isconjectured that the statistical properties of pseudointegrable billiards are intermediate between Poisson

statistics, typically related to integrable systems 12 , and the statistics of the Gaussian orthogonal ensemble 13 , related to fully chaotic time reversal invariant systems. The socalled ‘‘intermediate statistics’’ has also been observed in disordered, mesoscopic systems at themetal-insulator transition 14 , for systems with interacting electrons 15 , and for incommensurate double-walled carbon nanotubes 16 . A suitable measure for intermediate spectral statistics is 2 the level compressibility limL→ (L)/L, where 2 (L) is the number variance 13 for energy intervals of length L measured in units of the average level spacing . Note that coincides with the value of the spectral...