Gregory Berkolaiko

Star graphs and Seba billiards

We derive an exact expression for the two-point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity. This turns out to be identical to the corresponding result for certain Seba billiards in the semiclassical limit. Reasons for this are discussed. The formula we derive is also shown to be equivalent to a series expansion for the form factor - the Fourier transform of the two-point correlation function - previously calculated using periodic orbit theory.We derive an exact expression for the two-point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity. This turns out to be identical to the corresponding result for certain eba billiards in the semiclassical limit. The reasons for this are discussed. The formula we derive is also shown to be equivalent to a series expansion for the form factor - the Fourier transform of the two-point correlation function - previously calculated using periodic orbit theory.

Two-point spectral correlations for star graphs

The eigenvalues of the Schrodinger operator on a graph G are related via an exact trace formula to periodic orbits on G. This connection is used to calculate two-point spectral statistics for a particular family of graphs, called star graphs, in the limit as the number of edges tends to infinity. Combinatorial techniques are used to evaluate both the diagonal (same orbit) and off-diagonal (different orbit) contributions to the sum over pairs of orbits involved. In this way, a general formula is derived for terms in the (short-time) expansion of the form factor K() in powers of , and the first few are computed explicitly. The result demonstrates that K() is neither Poissonian nor random matrix, but an intermediate between the two. Off-diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients.

No Quantum Ergodicity for Star Graphs

We investigate statistical properties of the eigenfunctions of the Schrödinger operator on families of star graphs with incommensurate bond lengths. We show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average. We further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds. We describe how to construct such subsequences explicitly. These structures are analogous to scars on short unstable periodic orbits.

NON-EXPONENTIAL STABILITY AND DECAY RATES IN NONLINEAR STOCHASTIC DIFFERENCE EQUATION WITH UNBOUNDED NOISES

We consider the stochastic difference equation where f and g are nonlinear, bounded functions, is a sequence of independent random variables, and h>0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution . We also show that, for some natural choices of f and g, the rate of decay of is approximately polynomial: there exists such that decays faster than but slower than , for any . It turns out that, if decays faster than as , the polynomial rate of decay can be established precisely: tends to a constant limit. On the other hand, if g does not decay quickly enough, the approximate decay rate is the best possible result.

On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D

We study the spectral stability of solitary wave solutions to the nonlinear Dirac e quation in one dimension. We focus on the Dirac equation with cubic nonlinearity, known as the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model. Presented numerical computations of the spectrum of linearization at a solitary wave show that the solitary waves are spectrally stable. We corroborate our results by finding explicit expressions for several of the eigenfunction s. Some of the analytic results hold for the nonlinear Dirac equation with generic nonlinearity.

The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions

Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains νn of the n-th eigenfunction satisfies n ≥ νn. Here, we provide a new interpretation for the Courant nodal deficiency dn = n − νn in the case of quantum graphs. It equals the Morse index — at a critical point — of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaningit is the number of unstable directions in the vicinity of the critical point corresponding to the n-th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.

Form factor for a family of quantum graphs: an expansion to third order

For certain types of quantum graphs we show that the random matrix form factor can be recovered to at least third order in the scaled time τ from periodic-orbit theory. We consider the contributions from pairs of periodic orbits represented by diagrams with up to two self-intersections connected by up to four arcs and explain why all other diagrams are expected to give higher-order corrections only. For a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time-reversal symmetry sum to zero. The mechanism for this cancellation is rather general which suggests that it also applies at higher orders in the expansion. This expectation is in full agreement with the fact that in this case the linear-τ contribution, the diagonal approximation, already reproduces the random matrix form factor for τ < 1. For systems with time-reversal symmetry there are more diagrams which contribute at third order. We sum these contributions for quantum graphs with uniformly hyperbolic dynamics, obtaining +2τ3, in agreement with random matrix theory. As in the previous calculation of the leading-order correction to the diagonal approximation we find that the third-order contribution can be attributed to exceptional orbits representing the intersection of diagram classes.

Leading off-diagonal correction to the form factor of large graphs

Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, K(tau), of a generic quantum graph with mixing classical dynamics and time-reversal symmetry. We calculate the contribution from pairs of self-intersecting orbits that differ from each other only in the orientation of a single loop. In the limit of large graphs, these pairs produce a contribution -2tau(2) to the form factor which agrees with random-matrix theory.

Moments of the Wigner delay times

The Wigner time delay is a measure of the time spent by a particle inside the scattering region of an open system. For chaotic systems, the statistics of the individual delay times (whose average is the Wigner time delay) are thought to be well described by random matrix theory. Here we present a semiclassical derivation showing the validity of random matrix results. In order to simplify the semiclassical treatment, we express the moments of the delay times in terms of correlation functions of scattering matrices at different energies. In the semiclassical approximation, the elements of the scattering matrix are given in terms of the classical scattering trajectories, requiring one to study correlations between sets of such trajectories. We describe the structure of correlated sets of trajectories and formulate the rules for their evaluation to the leading order in inverse channel number. This allows us to derive a polynomial equation satisfied by the generating function of the moments. Along with showing the agreement of our semiclassical results with the moments predicted by random matrix theory, we infer that the scattering matrix is unitary to all orders in the semiclassical approximation.

Transport moments beyond the leading order

For chaotic cavities with scattering leads attached, transport properties can be approximated in terms of the classical trajectories that enter and exit the system. With a semiclassical treatment involving fine correlations between such trajectories, we develop a diagrammatic technique to calculate the moments of various transport quantities. Namely, we find the moments of the transmission and reflection eigenvalues for systems with and without time-reversal symmetry. We also derive related quantities involving an energy dependence: the moments of the Wigner delay times and the density of states of chaotic Andreev billiards, where we find that the gap in the density persists when subleading corrections are included. Finally, we show how to adapt our techniques to nonlinear statistics by calculating the correlation between transport moments. In each setting, the answer for the nth moment is obtained for arbitrary n (in the form of a moment generating function) and for up to three leading orders in terms of the inverse channel number. Our results suggest patterns that should hold for further corrections, and by matching with the lower-order moments available from random matrix theory, we derive the likely higher-order generating functions.