61 research outputs found
Exact, convergent periodic-orbit expansions of individual energy eigenvalues of regular quantum graphs
We present exact, explicit, convergent periodic-orbit expansions for
individual energy levels of regular quantum graphs. One simple application is
the energy levels of a particle in a piecewise constant potential. Since the
classical ray trajectories (including ray splitting) in such systems are
strongly chaotic, this result provides the first explicit quantization of a
classically chaotic system.Comment: 25 pages, 5 figure
Combinatorial identities for binary necklaces from exact ray-splitting trace formulae
Based on an exact trace formula for a one-dimensional ray-splitting system,
we derive novel combinatorial identities for cyclic binary sequences (P\'olya
necklaces).Comment: 15 page
Explicitly solvable cases of one-dimensional quantum chaos
We identify a set of quantum graphs with unique and precisely defined
spectral properties called {\it regular quantum graphs}. Although chaotic in
their classical limit with positive topological entropy, regular quantum graphs
are explicitly solvable. The proof is constructive: we present exact periodic
orbit expansions for individual energy levels, thus obtaining an analytical
solution for the spectrum of regular quantum graphs that is complete, explicit
and exact
A topological paradigm for hippocampal spatial map formation using persistent homology
Extent: 14 p.An animal's ability to navigate through space rests on its ability to create a mental map of its environment. The hippocampus is the brain region centrally responsible for such maps, and it has been assumed to encode geometric information (distances, angles). Given, however, that hippocampal output consists of patterns of spiking across many neurons, and downstream regions must be able to translate those patterns into accurate information about an animal's spatial environment, we hypothesized that 1) the temporal pattern of neuronal firing, particularly co-firing, is key to decoding spatial information, and 2) since co-firing implies spatial overlap of place fields, a map encoded by co-firing will be based on connectivity and adjacency, i.e., it will be a topological map. Here we test this topological hypothesis with a simple model of hippocampal activity, varying three parameters (firing rate, place field size, and number of neurons) in computer simulations of rat trajectories in three topologically and geometrically distinct test environments. Using a computational algorithm based on recently developed tools from Persistent Homology theory in the field of algebraic topology, we find that the patterns of neuronal co-firing can, in fact, convey topological information about the environment in a biologically realistic length of time. Furthermore, our simulations reveal a “learning region” that highlights the interplay between the parameters in combining to produce hippocampal states that are more or less adept at map formation. For example, within the learning region a lower number of neurons firing can be compensated by adjustments in firing rate or place field size, but beyond a certain point map formation begins to fail. We propose that this learning region provides a coherent theoretical lens through which to view conditions that impair spatial learning by altering place cell firing rates or spatial specificity.Y. Dabaghian, F. Mémoli, L. Frank, G. Carlsso
Spectra of regular quantum graphs
We consider a class of simple quasi one-dimensional classically
non-integrable systems which capture the essence of the periodic orbit
structure of general hyperbolic nonintegrable dynamical systems. Their behavior
is simple enough to allow a detailed investigation of both classical and
quantum regimes. Despite their classical chaoticity, these systems exhibit a
``nonintegrable analog'' of the Einstein-Brillouin-Keller quantization formula
which provides their spectra explicitly, state by state, by means of convergent
periodic orbit expansions.Comment: 32 pages, 10 figure
Weyl formulas for annular ray-splitting billiards
We consider the distribution of eigenvalues for the wave equation in annular
(electromagnetic or acoustic) ray-splitting billiards. These systems are
interesting in that the derivation of the associated smoothed spectral counting
function can be considered as a canonical problem. This is achieved by
extending a formalism developed by Berry and Howls for ordinary (without
ray-splitting) billiards. Our results are confirmed by numerical computations
and permit us to infer a set of rules useful in order to obtain Weyl formulas
for more general ray-splitting billiards
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