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arXiv:chao-dyn/9702006 · European Journal of Physics

The elliptic billiard: subtleties of separability

Ramses van Zon, Th.W. Ruijgrok

Published Jan 1, 1998• 8 citations
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Abstract

Some of the subtleties of the integrability of the elliptic quantum billiard are discussed. Considering a well known classical constant of the motion in the quantum case, we find that a naive calculation of the commutator with the Hamiltonian does not show whether or not it is zero. It is shown how this problem can be solved. A geometric picture is given that reveals why levels of a separable system cross. It is shown that the repulsions found by Ayant and Arvieu are computational effects and that the method used by Traiber et al is related to the present picture which explains the crossings they find. An asymptotic formula for the energy levels is derived and it is found that the statistical quantities of the spectrum P ( s ) and have the form expected for an integrable system.

DOI
10.1088/0143-0807/19/1/011

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Some of the subtleties of the integrability of the elliptic quantum billiard are discussed. Considering a well known classical constant of the motion in the quantum case, we find that a naive calculation of the commutator with the Hamiltonian does not show whether or not it is zero. It is shown how this problem can be solved. A geometric picture is given that reveals why levels of a separable system cross. It is shown that the repulsions found by Ayant and Arvieu are computational effects and that the method used by Traiber et al is related to the present picture which explains the crossings they find. An asymptotic formula for the energy levels is derived and it is found that the statistical quantities of the spectrum P ( s ) and have the form expected for an integrable system.
Some of the subtleties of the integrability of the elliptic quantum billiard are discussed. A well known classical constant of the motion has in the quantum case an ill-defined commutator with the Hamiltonian. It is shown how this problem can be solved. A geometric picture is given revealing why levels of a separable system cross. It is shown that the repulsions found by Ayant and Arvieu are computational effects and that the method used by Traiber et al is related to the present picture which explains the crossings they find. An asymptotic formula for the energy-levels is derived and it is found that the statistical quantities of the spectrum P(s) and ¯∆(L) have the form expected for an integrable system. Quelques-unes des subtilités de l’intégrabilité du billard elliptique quantique sont discutées. Dans le cas quantique, le commutateur avec l’Hamiltonien d’une certaine quantité connue pour ˜etre une constante du mouvement en méchanique classique, s’avère mal défini. On démontre ici comment on peut résoudre ce problème. Pour expliquer que les niveaux d’énergie d’un sytème séparable se croisent, une représentation géométrique est utilisée. On montre que les répulsions trouvées par Ayant et Arvieu sont des effets numériques et que la méthode employée par Traiber et al est liée à la représentation géométrique considérée ici, ce qui explique qu’ils trouvent aussi un croisement des niveaux d’énergie. Une expression asymptotique est derivée pour les niveaux d’énergie et les quantités statistiques du spectre P(s) et ¯ ∆(L) sont obtenues. Ils ont la forme prédite pour des systèmes intégrables.
The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schrödinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry-Tabor trace formula, resulting in a summation over non-periodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the The semiclassical quantization of a Hamiltonian system is deeply connected to the structure of its phase space. The generic Hamiltonian system contains a complicated mixture of near-integrable and
The subject of the four manuscripts which make up this dissertation is the concept of integrability in quantum mechanical systems with two degrees of freedom, including the relation between quantum integrability and classical integrability in such systems. In the first manuscript, I consider a two-spin model both classically and quantum mechanically. In the six-dimensional parameter space of this model, classical integrability is satisfied on a five-dimensional hypersurface, and level crossings occur on four-dimensional manifolds that are completely embedded in the integrability hypersurface except for some lower-dimensional sub-manifolds. Under mild assumptions, the classical integrability condition can be reconstructed from a purely quantum mechanical study of level degeneracies in finite-dimensional invariant blocks of the Hamiltonian matrix. In the second manuscript, I analyze the functional dependence [special characters omitted] of the Hamiltonian operator on the action operators for an integrable quantum system, and I compare it with the corresponding functional relationship H (p1, q1; p2, q 2) = HC(J1, J2) in the classical limit of that system. It is demonstrated that the functional dependence is not the same quantum mechanically and (semi-)classically. The former is shown to converge toward the latter in some asymptotic regime associated with the classical limit, but the convergence may be non-uniform. This analysis is carried out for an integrable one- parameter two-spin model. Additional results presented for the (integrable) circular billiard model exhibit similar quantum corrections to the predictions of semiclassical quantization. In the third and the fourth manuscripts, I investigate the relations between symmetry, integrability, and the assignment of quantum numbers to eigenstates for two-spin and spin-boson models. I calculate quantum invariants in the form of expectation values for selected operators and monitor their dependence on the Hamiltonian parameters along loops within, without, and across the integrability region in parameter space. I find clear-cut signatures of integrability and nonintegrability in the observed traces of quantum invariants evaluated in invariant Hilbert subspaces. The results support the notion that quantum integrability depends on the existence of action operators as constituent elements of the Hamiltonian.
We study the behaviour of the normal derivative of eigenfunctions of the Helmholtz equation inside billiards with Dirichlet boundary condition. These boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest. Therefore, they form a reduced representation of the quantum system, analogous to the Poincaré section of the classical system. For the normal derivatives we introduce an equivalent to the standard Green function and derive an integral equation on the boundary. Based on this integral equation we compute the first two terms of the mean asymptotic behaviour of the boundary functions for large energies. The first term is universal and independent of the shape of the billiard. The second one is proportional to the curvature of the boundary. The asymptotic behaviour is compared with numerical results for the stadium billiard, different limaçon billiards and the circle billiard, and good agreement is found. Furthermore, we derive an asymptotic completeness relation for the boundary functions.
We investigate the connection between energy level crossings in integrable systems and their integrability, i.e. the existence of a set of non-trivial integrals of motion. In particular, we consider a general quantum Hamiltonian linear in the coupling u , H ( u ) = T + uV , and require that it has the maximum possible number of nontrivial commuting partners also linear in u . We demonstrate how this commutation requirement alone leads to: (1) an exact solution for the energy spectrum and (2) level crossings, which are always present in these Hamiltonians in violation of the Wigner–von Neumann non-crossing rule. Moreover, we construct these Hamiltonians explicitly by resolving the above commutation requirement and show their equivalence to a sector of Gaudin magnets (central spin Hamiltonians). In contrast, fewer than the maximum number of conservation laws does not guarantee level crossings.