Abstract: We prove the spectral instability of the complex cubic oscillator
\({-\frac{{\rm d}^{2}}{{\rm d}x^{2}} + ix^{3} + i \alpha x}\)
for non-negative values of the parameter α, by getting the exponential growth rate of
\({\ \Pi_{n}(\alpha)\ }\)
, where
\({\Pi_{n}(\alpha)}\)
is the spectral projection associated with the nth eigenvalue of the operator. More precisely, we show that for all non-negative α
$$\lim\limits_{n \to + \infty} \frac{1}{n} {\rm log}\ \Pi_{n}(\alpha)\ = \frac{\pi}{\sqrt{3}}$$
. PubDate: 2014-10-01

Abstract: The Penrose inequality in Minkowski is a geometric inequality relating the total outer null expansion and the area of closed, connected and spacelike codimension-two surfaces
\({{\bf \mathcal{S}}}\)
in the Minkowski spacetime, subject to an additional convexity assumption. In a recent paper, Brendle and Wang A (Gibbons–Penrose inequality for surfaces in Schwarzschild Spacetime. arXiv:1303.1863, 2013) find a sufficient condition for the validity of this Penrose inequality in terms of the geometry of the orthogonal projection of
\({{\bf \mathcal{S}}}\)
onto a constant time hyperplane. In this work, we study the geometry of hypersurfaces in n-dimensional Euclidean space which are normal graphs over other surfaces and relate the intrinsic and extrinsic geometry of the graph with that of the base hypersurface. These results are used to rewrite Brendle and Wang’s condition explicitly in terms of the time height function of
\({{\bf \mathcal{S}}}\)
over a hyperplane and the geometry of the projection of
\({{\bf \mathcal{S}}}\)
along its past null cone onto this hyperplane. We also include, in Appendix, a self-contained summary of known and new results on the geometry of projections along the Killing direction of codimension two-spacelike surfaces in a strictly static spacetime. PubDate: 2014-10-01

Abstract: We show how to relate the full quantum dynamics of a spin-½ particle on
\({\mathbb{R}^d}\)
to a classical Hamiltonian dynamics on the enlarged phase space
\({\mathbb{R}^{2d} \times \mathbb{S}^{2}}\)
up to errors of second order in the semiclassical parameter. This is done via an Egorov-type theorem for normal Wigner–Weyl calculus for
\({\mathbb{R}^d}\)
(Folland, Harmonic Analysis on Phase Space, 1989; Lein, Weyl Quantization and Semiclassics, 2010) combined with the Stratonovich–Weyl calculus for SU(2) (Varilly and Gracia-Bondia, Ann Phys 190:107–148, 1989). For a specific class of Hamiltonians, including the Rabi- and Jaynes–Cummings model, we prove an Egorov theorem for times much longer than the semiclassical time scale. We illustrate the approach for a simple model of the Stern–Gerlach experiment. PubDate: 2014-10-01

Abstract: The interplay among the spectrum, geometry and magnetic field in tubular neighbourhoods of curves in Euclidean spaces is investigated in the limit when the cross section shrinks to a point. Proving a norm resolvent convergence, we derive effective, lower-dimensional models which depend on the intensity of the magnetic field and curvatures. The results are used to establish complete asymptotic expansions for eigenvalues. Spectral stability properties based on Hardy-type inequalities induced by magnetic fields are also analysed. PubDate: 2014-10-01

Abstract: We consider distributions on
\({\mathbb{R}^{n}{\setminus}\{0\}}\)
which satisfy a given set of partial differential equations and provide criteria for the existence of extensions to
\({\mathbb{R}^n}\)
that satisfy the same set of equations on
\({\mathbb{R}^n}\)
. We use the results to construct distributions satisfying specific renormalisation conditions in the Epstein and Glaser approach to perturbative quantum field theory. Contrary to other approaches, we provide a unified approach to treat Lorentz covariance, invariance under global gauge group and almost homogeneity, as well as discrete symmetries. We show that all such symmetries can be recovered by applying a linear map defined for all degrees of divergence. Using similar techniques, we find a relation between on-shell and off-shell time-ordered products involving higher derivatives of the fields. PubDate: 2014-10-01

Abstract: We construct a matrix model that reproduces the topological string partition function on arbitrary toric Calabi–Yau threefolds. This demonstrates, in accord with the BKMP “remodeling the B-model” conjecture, that Gromov–Witten invariants of any toric Calabi–Yau threefold can be computed in terms of the spectral invariants of a spectral curve. Moreover, it proves that the generating function of Gromov–Witten invariants is a tau function for an integrable hierarchy. In a follow-up paper, we will explicitly construct the spectral curve of our matrix model and argue that it equals the mirror curve of the toric Calabi–Yau manifold. PubDate: 2014-10-01

Abstract: We show that the critical Kac–Ward operator on isoradial graphs acts in a certain sense as the operator of s-holomorphicity, and we identify the fermionic observable for the spin Ising model as the inverse of this operator. This result is partially a consequence of a more general observation that the inverse Kac–Ward operator on any planar graph is given by what we call a fermionic generating function. We also present a general picture of the non-backtracking walk representation of the critical and supercritical inverse Kac–Ward operators on isoradial graphs. PubDate: 2014-10-01

Abstract: Consider a quantum dot coupled to two semi-infinite one-dimensional leads at thermal equilibrium. We turn on adiabatically a bias between the leads such that there exists exactly one discrete eigenvalue both at the beginning and at the end of the switching procedure. For a specific adiabatic limit scenario, it is shown that the expectation in the final bound state strongly depends on the history of the switching procedure. On the contrary, the contribution to the final steady-state corresponding to the continuous spectrum has no memory, and only depends on the initial and final values of the bias. PubDate: 2014-10-01

Abstract: We prove exponential decay for the solution of the Schrödinger equation on a dissipative waveguide. The absorption is effective everywhere on the boundary, but the geometric control condition is not satisfied. The proof relies on separation of variables and the Riesz basis property for the eigenfunctions of the transverse operator. The case where the absorption index takes negative values is also discussed. PubDate: 2014-09-16

Abstract: I study a Lindblad dynamics modeling a quantum test particle in a Dirac comb that collides with particles from a background gas. The main result is a homogenization theorem in an adiabatic limiting regime involving large initial momentum for the test particle. Over the time interval considered, the particle would exhibit essentially ballistic motion if either the singular periodic potential or the kicks from the gas were removed. However, the particle behaves diffusively when both sources of forcing are present. The conversion of the motion from ballistic to diffusive is generated by occasional quantum reflections that result when the test particle’s momentum is driven through a collision near to an element of the half-spaced reciprocal lattice of the Dirac comb. PubDate: 2014-09-16

Abstract: We study the invariant of knots in lens spaces defined from quantum Chern–Simons theory. By means of the knot operator formalism, we derive a generalization of the Rosso-Jones formula for torus knots in L(p,1). In the second part of the paper, we propose a B-model topological string theory description of torus knots in L(2,1). PubDate: 2014-09-16

Abstract: Using black hole inequalities and the increase of the horizon’s areas, we show that there are arbitrarily small electro-vacuum perturbations of the standard initial data of the extreme Reissner–Nordström black hole that (by contradiction) cannot decay in time into any extreme Kerr–Newman black hole. This proves that, in a formal sense, the reduced family of the extreme Kerr–Newman black holes is unstable. It remains of course to be seen whether the whole family of charged black holes, including those extremes, is stable or not. PubDate: 2014-09-16

Abstract: We revisit the problem of semiclassical spectral asymptotics for a pure magnetic Schrödinger operator on a two-dimensional Riemannian manifold. We suppose that the minimal value b
0 of the intensity of the magnetic field is strictly positive, and the corresponding minimum is unique and non-degenerate. The purpose is to get the control on the spectrum in an interval
\({(hb_0, h(b_0 + \gamma_0)]}\)
for some
\({\gamma_0 > 0}\)
independent of the semiclassical parameter h. The previous papers by Helffer–Mohamed and by Helffer–Kordyukov were only treating the ground-state energy or a finite (independent of h) number of eigenvalues. Note also that N. Raymond and S. Vũ Ngọc have recently developed a different approach of the same problem. PubDate: 2014-09-14

Abstract: We prove that in a certain class of conformal data on a manifold with ends of cylindrical type, if the conformally decomposed Einstein constraint equations do not admit a solution, then one can always find a nontrivial solution to the limit equation first explored by Dahl et al. (Duke Math J 161(14):2669–2798, 2012). We also give an example of a Ricci curvature condition on the manifold which precludes the existence of a solution to this limit equation. This shows that the limit equation criterion can be a useful tool for proving the existence of solutions to the Einstein constraint equations on manifolds with ends of cylindrical type. PubDate: 2014-09-14

Abstract: We construct an elementary, combinatorial kind of topological quantum field theory (TQFT), based on curves, surfaces, and orientations. The construction derives from contact invariants in sutured Floer homology and is essentially an elaboration of a TQFT defined by Honda–Kazez–Matić. This topological field theory stores information in binary format on a surface and has “digital” creation and annihilation operators, giving a toy-model embodiment of “it from bit”. PubDate: 2014-09-01

Abstract: We consider globally hyperbolic flat spacetimes in 2 + 1 and 3 + 1 dimensions, in which a uniform light signal is emitted on the r-level surface of the cosmological time for r → 0. We show that the frequency shift of this signal, as perceived by a fixed observer, is a well-defined, bounded function which is generally not continuous. This defines a model with anisotropic background radiation that contains information about initial singularity of the spacetime. In dimension 2 + 1, we show that this observed frequency shift function is stable under suitable perturbations of the spacetime, and that, under certain conditions, it contains sufficient information to recover its geometry and topology. We compute an approximation of this frequency shift function for a few simple examples. PubDate: 2014-09-01

Abstract: To study concentration and oscillation properties of eigenfunctions of the discrete Laplacian on regular graphs, we construct in this paper a pseudo-differential calculus on homogeneous trees, their universal covers. We define symbol classes and associated operators. We prove that these operators are bounded on L
2 and give adjoint and product formulas. Finally, we compute the symbol of the commutator of a pseudo-differential operator with the Laplacian. PubDate: 2014-09-01

Abstract: The overlap,
\({\mathcal{D}_N}\)
, between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is
\({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\)
with the so-called Anderson integral
\({\mathcal{I}_N}\)
. We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit
\({\mathcal{I}_N = \gamma\ln N + O(1)}\)
as
\({N\to\infty}\)
. The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on
\({\mathcal{D}_N}\)
concluding that
\({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\)
with constants C,
\({\tilde{C}}\)
, and
\({\tilde{\gamma}}\)
. In particular,
\({\mathcal{D}_N\to 0}\)
as
\({N\to\infty}\)
which is known as Anderson’s orthogonality catastrophe. PubDate: 2014-09-01

Abstract: Karl Löwner (later known as Charles Loewner) introduced his famous differential equation in 1923 to solve the Bieberbach conjecture for series expansion coefficients of univalent analytic functions at level n = 3. His method was revived in 1999 by Oded Schramm when he introduced the Stochastic Loewner Evolution (SLE), a conformally invariant process which made it possible to prove many predictions from conformal field theory for critical planar models in statistical mechanics. The aim of this paper is to revisit the Bieberbach conjecture in the framework of SLE processes and, more generally, Lévy processes. The study of their unbounded whole-plane versions leads to a discrete series of exact results for the expectations of coefficients and their variances, and, more generally, for the derivative moments of some prescribed order p. These results are generalized to the “oddified” or m-fold conformal maps of whole-plane SLEs or Lévy–Loewner Evolutions. We also study the (average) integral means multifractal spectra of these unbounded whole-plane SLE curves. We prove the existence of a phase transition at a moment order p = p
*(κ) > 0, at which one goes from the bulk SLE
κ
average integral means spectrum, as predicted by the first author (Duplantier Phys. Rev. Lett. 84:1363–1367, 2000) and established by Beliaev and Smirnov (Commun Math Phys 290:577–595, 2009) and valid for p ≤ p
*(κ), to a new integral means spectrum for p ≥ p
*(κ), as conjectured in part by Loutsenko (J Phys A Math Gen 45(26):265001, 2012). The latter spectrum is, furthermore, shown to be intimately related, via the associated packing spectrum, to the radial SLE derivative exponents obtained by Lawler, Schramm and Werner (Acta Math 187(2):237–273, 2001), and to the local SLE tip multifractal exponents obtained from quantum gravity by the first author (Duplantier Proc. Sympos. Pure Math. 72(2):365–482, 2004). This is generalized to the integral means spectrum of the m-fold transform of the unbounded whole-plane SLE map. A succinct, preliminary, version of this study first appeared in Duplantier et al. (Coefficient estimates for whole-plane SLE processes, Hal-00609774, 2011). PubDate: 2014-08-28

Abstract: In this paper, we study the Weyl symbol of the Schrödinger semigroup e−tH
, H = −Δ + V, t > 0, on
\({L^2(\mathbb{R}^n)}\)
, with nonnegative potentials V in
\({L^1_{\rm loc}}\)
. Some general estimates like the L
∞ norm concerning the symbol u are derived. In the case of large dimension, typically for nearest neighbor or mean field interaction potentials, we prove estimates with parameters independent of the dimension for the derivatives
\({\partial_x^\alpha\partial_\xi^\beta u}\)
. In particular, this implies that the symbol of the Schrödinger semigroups belongs to the class of symbols introduced in Amour et al. (To appear in Proceedings of the AMS) in a high-dimensional setting. In addition, a commutator estimate concerning the semigroup is proved. PubDate: 2014-08-26