Topological calculation of the phase of the determinant of a non self-adjoint elliptic operator

We study the zeta-regularized determinant of a non self-adjoint elliptic operator on a closed odd-dimensional manifold. We show that, if the spectrum of the operator is symmetric with respect to the imaginary axis, then the determinant is real and its sign is determined by the parity of the number of the eigenvalues of the operator, which lie on the positive part of the imaginary axis. It follows that, for many geometrically defined operators, the phase of the determinant is a topological invariant. In numerous examples, coming from geometry and physics, we calculate the phase of the determinants in purely topological terms. Some of those examples were known in physical literature, but no mathematically rigorous proofs and no general theory were available until now.Comment: To appear in Communications of Mathematical Physic

Entanglement of Collectively Interacting Harmonic Chains

We study the ground-state entanglement of one-dimensional harmonic chains that are coupled to each other by a collective interaction as realized e.g. in an anisotropic ion crystal. Due to the collective type of coupling, where each chain interacts with every other one in the same way,the total system shows critical behavior in the direction orthogonal to the chains while the isolated harmonic chains can be gapped and non-critical. We derive lower and most importantly upper bounds for the entanglement,quantified by the von Neumann entropy, between a compact block of oscillators and its environment. For sufficiently large size of the subsystems the bounds coincide and show that the area law for entanglement is violated by a logarithmic correction.Comment: 5 pages, 1 figur

Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix

The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a matrix $X$ with columns that form an orthonormal basis for a subspace \X, and a Hermitian matrix $A$, the eigenvalues of $X^HAX$ are called Ritz values of $A$ with respect to \X. If the subspace \X is $A$-invariant then the Ritz values are some of the eigenvalues of $A$. If the $A$-invariant subspace \X is perturbed to give rise to another subspace \Y, then the vector of absolute values of changes in Ritz values of $A$ represents the absolute eigenvalue approximation error using \Y. We bound the error in terms of principal angles between \X and \Y. We capitalize on ideas from a recent paper [DOI: 10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute values of differences between Ritz values for subspaces \X and \Y was weakly (sub-)majorized by a constant times the sine of the vector of principal angles between \X and \Y, the constant being the spread of the spectrum of $A$. In that result no assumption was made on either subspace being $A$-invariant. It was conjectured there that if one of the trial subspaces is $A$-invariant then an analogous weak majorization bound should only involve terms of the order of sine squared. Here we confirm this conjecture. Specifically we prove that the absolute eigenvalue error is weakly majorized by a constant times the sine squared of the vector of principal angles between the subspaces \X and \Y, where the constant is proportional to the spread of the spectrum of $A$. For many practical cases we show that the proportionality factor is simply one, and that this bound is sharp. For the general case we can only prove the result with a slightly larger constant, which we believe is artificial.Comment: 12 pages. Accepted to SIAM Journal on Matrix Analysis and Applications (SIMAX

Periodic solutions for completely resonant nonlinear wave equations

We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=\phi(u)$, with $\phi(u)=\Phi u^{3} + O(u^{5})$ odd and analytic, $\Phi\neq0$, and we construct small amplitude periodic solutions with frequency \o for a large Lebesgue measure set of \o close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and nonresonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect the nonlinear wave equations $u_{xx}-u_{tt}+ M u = \phi(u)$, $M\neq0$, is that not only the P equation but also the Q equation is infinite-dimensiona

Quantum state transformations and the Schubert calculus

Recent developments in mathematics have provided powerful tools for comparing the eigenvalues of matrices related to each other via a moment map. In this paper we survey some of the more concrete aspects of the approach with a particular focus on applications to quantum information theory. After discussing the connection between Horn's Problem and Nielsen's Theorem, we move on to characterizing the eigenvalues of the partial trace of a matrix.Comment: 40 pages. Accepted for publication in Annals of Physic

The Lagrange Equilibrium Points L_4 and L_5 in a Black Hole Binary System

We calculate the location and stability of the L_4 and L_5 Lagrange equilibrium points in the circular restricted three-body problem as the binary system evolves via gravitational radiation losses. Relative to the purely Newtonian case, we find that the L_4 equilibrium point moves towards the secondary mass and becomes slightly less stable, while the L_5 point moves away from the secondary and gains in stability. We discuss a number of astrophysical applications of these results, in particular as a mechanism for producing electromagnetic counterparts to gravitational-wave signals.Comment: 10 pages, 4 figures, submitted to ApJ; comments welcom

UV/Optical Detections of Candidate Tidal Disruption Events by GALEX and CFHTLS

We present two luminous UV/optical flares from the nuclei of apparently inactive early-type galaxies at z=0.37 and 0.33 that have the radiative properties of a flare from the tidal disruption of a star. In this paper we report the second candidate tidal disruption event discovery in the UV by the GALEX Deep Imaging Survey, and present simultaneous optical light curves from the CFHTLS Deep Imaging Survey for both UV flares. The first few months of the UV/optical light curves are well fitted with the canonical t^(-5/3) power-law decay predicted for emission from the fallback of debris from a tidally disrupted star. Chandra ACIS X-ray observations during the flares detect soft X-ray sources with T_bb= (2-5) x 10^5 K or Gamma > 3 and place limits on hard X-ray emission from an underlying AGN down to L_X (2-10 keV) <~ 10^41 ergs/s. Blackbody fits to the UV/optical spectral energy distributions of the flares indicate peak flare luminosities of > 10^44-10^45 ergs/s. The temperature, luminosity, and light curves of both flares are in excellent agreement with emission from a tidally disrupted main sequence star onto a central black hole of several times 10^7 msun. The observed detection rate of our search over ~ 2.9 deg^2 of GALEX Deep Imaging Survey data spanning from 2003 to 2007 is consistent with tidal disruption rates calculated from dynamical models, and we use these models to make predictions for the detection rates of the next generation of optical synoptic surveys.Comment: 28 pages, 27 figures, 11 tables, accepted to ApJ, final corrections from proofs adde

Discovery of an Ultrasoft X-ray Transient Source in the 2XMM Catalog: a Tidal Disruption Event Candidate

We have discovered an ultrasoft X-ray transient source, 2XMMi J184725.1-631724, which was detected serendipitously in two XMM-Newton observations in the direction of the center of the galaxy IC 4765-f01-1504 at a redshift of 0.0353. These two observations were separated by 211 days, with the 0.2-10 keV absorbed flux increasing by a factor of about 9. Their spectra are best described by a model dominated by a thermal disk or a single-temperature blackbody component (contributing >80% of the flux) plus a weak power-law component. The thermal emission has a temperature of a few tens of eV, and the weak power-law component has a photon index of ~3.5. Similar to the black hole X-ray binaries in the thermal state, our source exhibits an accretion disk whose luminosity appears to follow the $L\propto T^4$ relation. This would indicate that the black hole mass is about 10^5-10^6 M_sun using the best-fitting inner disk radius. Both XMM-Newton observations show variability of about 21% on timescales of hours, which can be explained as due to fast variations in the mass accretion rate. The source was not detected by ROSAT in an observation in 1992, indicating a variability factor of >64 over longer timescales. The source was not detected again in X-rays in a Swift observation in 2011 February, implying a flux decrease by a factor of >12 since the last XMM-Newton observation. The transient nature, in addition to the extreme softness of the X-ray spectra and the inactivity of the galaxy implied by the lack of strong optical emission lines, makes it a candidate tidal disruption event. If this is the case, the first XMM-Newton observation would have been in the rising phase, and the second one in the decay phase.Comment: 12 pages, 6 figures. Accepted for publication in Ap

Lower bounds on the complexity of simulating quantum gates

We give a simple proof of a formula for the minimal time required to simulate a two-qubit unitary operation using a fixed two-qubit Hamiltonian together with fast local unitaries. We also note that a related lower bound holds for arbitrary n-qubit gates.Comment: 6 page

On one mechanism of light ablation of nanostructures

The mechanism of mechanical ablation of nanoparticles during the interaction with a high-power laser radiation pulse is proposed. A particle is polarized under a laser electric field, and electric forces acting on field-induced oppositesign charges cause rupture stresses. Upon reaching the stresses exceeding the maximum allowable values for a given material, a nanoparticle decays into two ones. This effect can be used for narrowing the size distribution of nanoparticles produced by the laser ablation method