Spectral Stability of the Neumann Laplacian

We prove the equivalence of Hardy- and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform H\"older category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount. AMS subject classifications: 35P15, 35J25, 47A75, 47B25, 26D10, 46E35. Keywords: Neumann Laplacian, Sobolev inequalities, Hardy inequalities, spectral stability, H\"older continuity.Comment: 23 page

Measurements continuous in time and a posteriori states in quantum

Measurements continuous in time were consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after the measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented, in the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that one in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.Comment: 31 page. See also related papers at http://www.maths.nott.ac.uk/personal/vpb/research/mes_fou.html and http://www.maths.nott.ac.uk/personal/vpb/research/fil_con.htm

The Propeller Regime of Disk Accretion to a Rapidly Rotating Magnetized Star

The propeller regime of disk accretion to a rapidly rotating magnetized star is investigated here for the first time by axisymmetric 2.5D magnetohydrodynamic simulations. An expanded, closed magnetosphere forms in which the magnetic field is predominantly toroidal. A smaller fraction of the star's poloidal magnetic flux inflates vertically, forming a magnetically dominated tower. Matter accumulates in the equatorial region outside magnetosphere and accretes to the star quasi-periodically through elongated funnel streams which cause the magnetic field to reconnect. The star spins-down owing to the interaction of the closed magnetosphere with the disk. For the considered conditions, the spin-down torque varies with the angular velocity of the star omega* as omega*^1.3 for fixed mass accretion rate. The propeller stage may be important in the evolution of X-ray pulsars, cataclysmic variables and young stars. In particular, it may explain the present slow rotation of the classical T Tauri stars.Comment: 5 pages with 4 figures, LaTeX, macros: emulapj.sty, avi movies are available at http://www.astro.cornell.edu/us-russia/disk_prop.ht

Ablation of solids by femtosecond lasers: ablation mechanism and ablation thresholds for metals and dielectrics

The mechanism of ablation of solids by intense femtosecond laser pulses is described in an explicit analytical form. It is shown that at high intensities when the ionization of the target material is complete before the end of the pulse, the ablation mechanism is the same for both metals and dielectrics. The physics of this new ablation regime involves ion acceleration in the electrostatic field caused by charge separation created by energetic electrons escaping from the target. The formulae for ablation thresholds and ablation rates for metals and dielectrics, combining the laser and target parameters, are derived and compared to experimental data. The calculated dependence of the ablation thresholds on the pulse duration is in agreement with the experimental data in a femtosecond range, and it is linked to the dependence for nanosecond pulses.Comment: 27 pages incl.3 figs; presented at CLEO-Europe'2000 11-15 Sept.2000; papers QMD6 and CTuK11

Statistics of Neutron Stars at the Stage of Supersonic Propeller

We analyze the statistical distribution of neutron stars at the stage of a supersonic propeller. An important point of our analysis is allowance for the evolution of the angle of inclination of the magnetic axis to the spin axis of the neutron star for the boundary of the transition to the supersonic propeller stage for two models: the model with hindered particle escape from the stellar surface and the model with free particle escape. As a result, we have shown that a consistent allowance for the evolution of the inclination angle in the region of extinct radio pulsars for the two models leads to an increase in the total number of neutron stars at the supersonic propeller stage. This increase stems from he fact that when allowing for the evolution of the inclination angle $\chi$ for neutron stars in the region of extinct radio pulsars and, hence, for the boundary of the transition to the propeller stage, this transition is possible at shorter spin periods (P~5-10 s) than assumed in the standard model.Comment: 15 pages, 6 figures; scale corrected for figures 3-

Non-Weyl asymptotics for quantum graphs with general coupling conditions

Inspired by a recent result of Davies and Pushnitski, we study resonance asymptotics of quantum graphs with general coupling conditions at the vertices. We derive a criterion for the asymptotics to be of a non-Weyl character. We show that for balanced vertices with permutation-invariant couplings the asymptotics is non-Weyl only in case of Kirchhoff or anti-Kirchhoff conditions, while for graphs without permutation numerous examples of non-Weyl behaviour can be constructed. Furthermore, we present an insight helping to understand what makes the Kirchhoff/anti-Kirchhoff coupling particular from the resonance point of view. Finally, we demonstrate a generalization to quantum graphs with nonequal edge weights.Comment: minor changes, to appear in Pierre Duclos memorial issue of J. Phys. A: Math. Theo

Diffusion Approximation of Stochastic Master Equations with Jumps

In the presence of quantum measurements with direct photon detection the evolution of open quantum systems is usually described by stochastic master equations with jumps. Heuristically, from these equations one can obtain diffusion models as approximation. A necessary condition for a general diffusion approximation for jump master equations is presented. This approximation is rigorously proved by using techniques for Markov process which are based upon the convergence of Markov generators and martingale problems. This result is illustrated by rigorously obtaining the diffusion approximation for homodyne and heterodyne detection.Comment: 15 page

Dynamics of open quantum systems initially entangled with environment: Beyond the Kraus representation

We present a general analysis of the role of initial correlations between the open system and an environment on quantum dynamics of the open system.Comment: 5 revtex pages, no figures, accepted for publication in Phys. Rev.

Classical and nonclassical randomness in quantum measurements

The space of positive operator-valued measures on the Borel sets of a compact (or even locally compact) Hausdorff space with values in the algebra of linear operators acting on a d-dimensional Hilbert space is studied from the perspectives of classical and non-classical convexity through a transform $\Gamma$ that associates any positive operator-valued measure with a certain completely positive linear map of the homogeneous C*-algebra $C(X)\otimes B(H)$ into $B(H)$. This association is achieved by using an operator-valued integral in which non-classical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse $\Omega$ for $\Gamma$ yields an integral representation, along the lines of the classical Riesz Representation Theorem for certain linear functionals on $C(X)$, of certain (but not all) unital completely positive linear maps $\phi:C(X)\otimes B(H) \rightarrow B(H)$. The extremal and C*-extremal points of the space of POVMS are determined.Comment: to appear in Journal of Mathematical Physic

The 24-Cell and Calabi-Yau Threefolds with Hodge Numbers (1,1)

Calabi-Yau threefolds with h^11(X)=h^21(X)=1 are constructed as free quotients of a hypersurface in the ambient toric variety defined by the 24-cell. Their fundamental groups are SL(2,3), a semidirect product of Z_3 and Z_8, and Z_3 x Q_8.Comment: 22 pages, 3 figures, 3 table