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Register a new userIntermediate statistics for a system with symplectic symmetry: the Dirac rose graph
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We study the spectral statistics of the Dirac operator on a rose-shaped graph---a graph with a single vertex and all bonds connected at both ends to the vertex. We formulate a secular equation that generically determines the eigenvalues of the Dirac rose graph, which is seen to generalise the secular equation for a star graph with Neumann boundary conditions. We derive approximations to the spectral pair correlation function at large and small values of spectral spacings, in the limit as the number of bonds approaches infinity, and compare these predictions with results of numerical calculations. Our results represent the first example of intermediate statistics from the symplectic symmetry class.
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Jacobis method is a well-known algorithm in linear algebra to diagonalize symmetric matrices by successive elementary rotations. We report about the generalization of these elementary rotations towards canonical transformations acting in Hamiltonian phase spaces. This generalization allows to use Jacobis method in order to compute eigenvalues and eigenvectors of Hamiltonian (and skew-Hamiltonian) matrices with either purely real or purely imaginary eigenvalues by successive elementary symplectic decoupling-transformations.
We consider a three-dimensional chaotic system consisting of the suspension of Arnolds cat map coupled with a clock via a weak dissipative interaction. We show that the coupled system displays a synchronization phenomenon, in the sense that the relative phase between the suspension flow and the clock locks to a special value, thus making the motion fall onto a lower dimensional attractor. More specifically, we construct the attractive invariant manifold, of dimension smaller than three, using a convergent perturbative expansion. Moreover, we compute via convergent series the Lyapunov exponents, including notably the central one. The result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model. The main novelty of the current construction relies in the computation of the Lyapunov spectrum, which consists of non-trivial analytic exponents. Some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed.
The orbits and the dynamical symmetries for the screened Coulomb potentials and isotropic harmonic oscillators have been studied by Wu and Zeng [Z. B. Wu and J. Y. Zeng, Phys. Rev. A 62,032509 (2000)]. We find the similar properties in the responding systems in a spherical space, whose dynamical symmetries are described by Higgs Algebra. There exists a conserved aphelion and perihelion vector, which, together with angular momentum, constitute the generators of the geometrical symmetry group at the aphelia and perihelia points $(dot{r}=0)$.
We consider the dynamics of a quantum particle of mass $m$ on a $n$-edges star-graph with Hamiltonian $H_K=-(2m)^{-1}hbar^2 Delta$ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kreu{i}ns theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple $(H_K,H_{D}^{oplus})$, where $H_{D}^{oplus}$ is the free Hamiltonian with Dirichlet conditions in the vertex.
We consider a Dirac operator with a dislocation potential on the real line. The dislocation potential is a fixed periodic potential on the negative half-line and the same potential but shifted by real parameter $t$ on the positive half-line. Its spectrum has an absolutely continuous part (the union of bands separated by gaps) plus at most two eigenvalues in each non-empty gap. Its resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface. We prove that it has only two simple poles on each open gap: on the first sheet (an eigenvalue) or on the second sheet (a resonance). These poles are called states and there are no other poles. We prove: 1) each state is a continuous function of $t$, and we obtain its local asymptotic; 2) for each $t$ states in the gap are distinct; 3) in general, a state is non-monotone function of $t$ but it can be monotone for specific potentials; 4) we construct examples of operators, which have: a) one eigenvalue and one resonance in any finite number of gaps; b) two eigenvalues or two resonances in any finite number of gaps; c) two static virtual states in one gap.
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