Hyperbolic Trap Research Articles

Stationarity and Fredholm Theory in Subextremal Kerr-de Sitter Spacetimes

In a recent paper, we proved that solutions to linear wave equations in a subextremal Kerr-de Sitter spacetime have asymptotic expansions in quasinormal modes up to a decay order given by the normally hyperbolic trapping, extending the results of Vasy (2013). One central ingredient in the argument was a new definition of quasinormal modes, where a non-standard choice of stationary Killing vector field had to be used in order for the Fredholm theory to be applicable. In this paper, we show that there is in fact a variety of allowed choices of stationary Killing vector fields. In particular, the horizon Killing vector fields work for the analysis, in which case one of the corresponding ergoregions is completely removed.

Wave equations in the Kerr–de Sitter spacetime: The full subextremal range

We prove that solutions to linear wave equations in a subextremal Kerr–de Sitter spacetime have asymptotic expansions in quasinormal modes up to a decay order given by the normally hyperbolic trapping, extending the result of the second named author (2013). The main novelties are a different way of obtaining a Fredholm setup that defines the quasinormal modes and a new analysis of the trapping of lightlike geodesics in the Kerr–de Sitter spacetime, both of which apply in the full subextremal range. In particular, this reduces the question of decay for solutions to wave equations to the question of mode stability.

The analytical solution for the optimum voltage on regularizing electrodes of the open dynamically harmonized cell.

The Fourier transform ion cyclotron resonance method holds the lead in mass accuracy and resolving power among all other mass spectrometry methods. The dynamically harmonized cell is largely responsible for the supremacy. This cell has an ideal hyperbolic trapping potential after averaging over fast cyclotron motion. Recently we have introduced an open modification of the cell (especially useful with ultrahigh magnetic fields) and have found the analytical solution for the averaged potential inside it. The voltage on specific "regularizing" electrodes determines how close a potential is to the hyperbolic one. In this article, we find the optimal voltage on these "regularizing" electrodes analytically. This will assist with both further analysis and tuning of the trap after manufacturing.

Normally hyperbolic trapping on asymptotically stationary spacetimes

We prove microlocal estimates at the trapped set of asymptotically Kerr spacetimes: these are spacetimes whose metrics decay inverse polynomially in time to a stationary subextremal Kerr metric. This combines two independent results. The first one is purely dynamical: we show that the stable and unstable manifolds of a decaying perturbation of a time-translation-invariant dynamical system with normally hyperbolic trapping are smooth and decay to their stationary counterparts. The second, independent, result provides microlocal estimates for operators whose null-bicharacteristic flow has a normally hyperbolic invariant manifold, under suitable non-degeneracy conditions on the stable and unstable manifolds; this includes operators on closed manifolds, as well as operators on spacetimes for which the invariant manifold lies at future infinity.

High-frequency Bounds for the Helmholtz Equation Under Parabolic Trapping and Applications in Numerical Analysis

This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general “worst case” bound of exponential growth applies, and examples show that this growth can be realized through some sequence of wavenumbers; (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly convex obstacles (or several obstacles with additional constraints) the nontrapping resolvent estimate holds with a logarithmic loss. This paper proves the first resolvent estimate for parabolic trapping by obstacles, studying a class of obstacles the prototypical example of which is the exterior of two squares (in two dimensions) or two cubes (in three dimensions), whose sides are parallel. We show, via developments of the vector-field/multiplier argument of Morawetz and the first application of this methodology to trapping configurations, that a resolvent estimate holds with a polynomial loss over the nontrapping estimate. We use this bound, along with the other trapping resolvent estimates, to prove results about integral equation formulations of the boundary value problem in the case of trapping. Feeding these bounds into existing frameworks for analyzing finite and boundary element methods, we obtain the first wavenumber-explicit proofs of convergence for numerical methods for solving the Helmholtz equation in the exterior of a trapping obstacle.

Tomas–Stein Restriction Estimates on Convex Cocompact Hyperbolic Manifolds

Abstract In this paper, we investigate the Tomas–Stein restriction estimates on convex cocompact hyperbolic manifolds $\Gamma \backslash{\mathbb{H}}^{n+1}$. Via the spectral measure of the Laplacian, we prove that the Tomas–Stein restriction estimate holds when the limit set has Hausdorff dimension $\delta _\Gamma <n/2$. This provides an example for which restriction estimate holds in the presence of hyperbolic geodesic trapping.

Local Marked Boundary Rigidity Under Hyperbolic Trapping Assumptions

Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points, and a hyperbolic trapped set are locally marked boundary rigid.

The semiclassical resolvent on conformally compact manifolds with variable curvature at infinity

ABSTRACTWe construct a semiclassical parametrix for the resolvent of the Laplacian acting on functions on nontrapping conformally compact manifolds with variable sectional curvature at infinity. We apply this parametrix to analyze the Schwartz kernel of the semiclassical resolvent and Poisson operator and to show that the semiclassical scattering matrix is a semiclassical Fourier Integral Operator of appropriate class that quantizes the scattering relation. We also obtain high energy estimates for the resolvent and show existence of resonance free strips of arbitrary height away from the imaginary axis. We then use the results of Datchev and Vasy on gluing semiclassical resolvent estimates to obtain semiclassical resolvent estimates on certain conformally compact manifolds with hyperbolic trapping.

Spectral gaps for normally hyperbolic trapping

We establish a resonance free strip for codimension 2 symplectic normally hyperbolic trapped sets with smooth incoming/outgoing tails. An important application is wave decay on Kerr and Kerr–de Sitter black holes. We recover the optimal size of the strip and give an o(h -2 ) resolvent bound there. We next show existence of deeper resonance free strips under the r-normal hyperbolicity assumption and a pinching condition. We also give a lower bound on the one-sided cutoff resolvent on the real line.

Decay of correlations for normally hyperbolic trapping

We prove that for evolution problems with normally hyperbolic trapping in phase space, correlations decay exponentially in time. Normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it. Flows with this structure include contact Anosov flows, classical flows in molecular dynamics, and null geodesic flows for black holes metrics. The decay of correlations is a consequence of the existence of resonance free strips for Green’s functions (cut-off resolvents) and polynomial bounds on the growth of those functions in the semiclassical parameter.

Space charge and collective oscillation of ion cloud in a linear Paul trap

Charged particles confined in an ion trap is still the simplest system to study the effects like anharmonicity, parametric oscillation, bi-stability and in addition space charge induced modification to all the above effects. Some of these effects like space charge induced frequency shift and emergence of collective oscillation has been observed in a linear Paul trap. These results though similar to earlier observations made on a hyperbolic Paul trap, they differ in some finer details which has been addressed in this article. A simplistic model, limited to the adiabatic approximation, has been utilized to support the observation made in the experiment.

Non-trapping estimates near normally hyperbolic trapping

In this paper we prove semiclassical resolvent estimates for operators with normally hyperbolic trapping which are lossless relative to non-trapping estimates but take place in weaker function spaces. In particular, we obtain non-trapping estimates in standard $L^2$ spaces for the resolvent sandwiched between operators which localize away from the trapped set $\Gamma$ in a rather weak sense, namely whose principal symbols vanish on $\Gamma$.

Quasilinear waves and trapping: Kerr-de Sitter space

In these notes, we will describe recent work on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr-de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non-elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally hyperbolic trapping results of Dyatlov and a Nash-Moser iteration scheme.

Near Sharp Strichartz Estimates with Loss in the Presence of Degenerate Hyperbolic Trapping

We consider an n-dimensional spherically symmetric, asymptotically Euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic. By separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond Ehrenfest time, we show that solutions to the linear Schrödinger equation with initial conditions localized on a spherical harmonic satisfy Strichartz estimates with a loss depending only on the dimension n and independent of the degeneracy. The Strichartz estimates are sharp up to an arbitrary β > 0 loss. This is in contrast to Christianson and Wunsch (Amer J Math, 2013), where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set, independent of the dimension.

Near Sharp Strichartz Estimates with Loss in the Presence of Degenerate Hyperbolic Trapping

Near Sharp Strichartz Estimates with Loss in the Presence of Degenerate Hyperbolic Trapping

Experimental Determination (∼mHz) of the Ground-State Hyperfine Separation of Trapped 199Hg+ in a Hyperbolic Paul Trap

The Ramsey fringes on the ground-state hyperfine transition of 199Hg+ (ΔF = 1, ΔmF = 0) ions trapped in a hyperbolic Paul trap are firstly observed with the method of time-separated oscillatory fields. The full width at half maximum of central Ramsey fringes is about 25 mHz and the corresponding quality factor Q of the line is greater than 1012 for the trapped 199Hg+ microwave frequency standard. The hyperfine transition frequency shifted by magnetic field is also measured by the high-resolution Ramsey fringes. The final result is Δνh f s = 40507347997.3(0.5) Hz, which is corrected to zero magnetic field.

Optimization of a quadrupole ion storage trap as a source for time‐of‐flight mass spectrometry

Designs of a quadrupole ion trap (QIT) as a source for time-of-flight (TOF) mass spectrometry are evaluated for mass resolution, ion trapping, and laser activation of trapped ions. Comparisons are made with the standard hyperbolic electrode ion trap geometry for TOF mass analysis in both linear and reflectron modes. A parallel-plate design for the QIT is found to give significantly improved TOF mass spectrometer performance. Effects of ion temperature, trapped ion cloud size, mass, and extraction field on mass resolution are investigated in detail by simulation of the TOF peak profiles. Mass resolution (m/Δm) values of several thousand are predicted even at room temperature with moderate extraction fields for the optimized design. The optimized design also allows larger radial ion collection size compared with the hyperbolic ion trap, without compromising the mass resolution. The proposed design of the QIT also improves the ion-laser interaction volume and photon collection efficiency for fluorescence measurements on trapped ions.

A split-ring Paul trap for dipolar excitation of the radial ion motion and ellipticity studies

A radial dipolar excitation has been applied to a hyperbolic Paul trap with a segmented ring electrode. In addition, this configuration allowed the generation of an elliptical Paul trap by superimposing the RF trapping field with an electrostatic azimuthally quadrupolar field. The changes in the radial ion motions with respect to the conventional Paul trap are discussed and investigated experimentally. As expected, the formerly degenerated radial eigenfrequencies $\nu_{r}$ of the standard Paul trap split into two separate resonances at $\nu_{x}$ and $\nu_{y}$ , while the axial eigenfrequency $\nu_{z}$ remains unaffected. The new trap design adds flexibility and extends the range of applications of Paul traps.

Minoration de la résolvante dans le cas captif

In this note, we prove an optimal universal lower bound on the truncated resolvent for semiclassical Schrödinger operators near a trapping energy. In particular, this shows that known upper bounds for hyperbolic trapping are optimal. The proof rely on an idea of X.P. Wang, and on propagation of coherent states for Ehrenfest times.

A Quadrupole Ion Trap with Cylindrical Geometry Operated in the Mass-Selective Instability Mode

A cylindrical geometry ion trap is used to record mass spectra in the mass-selective instability mode. The geometry of the cylindrical ion trap has been optimized to maximize the quadrupole field component relative to the higher-order field content through field calculations using the Poisson/Superfish code and through experimental variation of the electrode structure. The results correspond well with predictions of the calculations. The trap has been used to record mass spectra with better than unit mass resolution, high sensitivity, and a mass/charge range of ∼600 Th. Multistage (MS(3)) experiments have been performed, and the Mathieu stability region has been experimentally mapped. The performance of this device compares satisfactorily with that of the hyperbolic ion trap.