Annales Henri Poincaré最新文献

We define and analyze the fermionic entanglement entropy of a Schwarzschild black hole horizon for the regularized vacuum state of an observer at infinity. Using separation of variables and an integral representation of the Dirac propagator, the entanglement entropy is computed to be a prefactor times the number of occupied angular momentum modes on the event horizon.

We revisit the Cauchy problem for the logarithmic Schrödinger equation and construct strong solutions in (H^1), the energy space, and the (H^2)-energy space. The solutions are provided in a constructive way, which does not rely on compactness arguments, that a sequence of approximate solutions forms a Cauchy sequence in a complete function space and then actual convergence is shown to be in a strong sense.

We study a two-state quantum system with a nonlinearity intended to describe interactions with a complex environment, arising through a nonlocal coupling term. We study the stability of particular solutions, obtained as constrained extrema of the energy functional of the system. The simplicity of the model allows us to justify a complete stability analysis. This is the opportunity to review in detail the techniques to investigate the stability issue. We also bring out the limitations of perturbative approaches based on simpler asymptotic models.

We consider essential self-adjointness on the space (C_0^{infty }((0,infty ))) of even-order, strongly singular, homogeneous differential operators associated with differential expressions of the type

in (L^2((0,infty );dx)). While the special case (n=1) is classical and it is well known that (tau _2(c)big |_{C_0^{infty }((0,infty ))}) is essentially self-adjoint if and only if (c ge 3/4), the case (n in {{mathbb {N}}}), (n ge 2), is far from obvious. In particular, it is not at all clear from the outset that

As one of the principal results of this paper we indeed establish the existence of (c_n), satisfying (c_n ge (4n-1)!!big /2^{2n}), such that property (*) holds. In sharp contrast to the analogous lower semiboundedness question,

which permits the sharp (and explicit) answer (c ge [(2n -1)!!]^{2}big /2^{2n}), (n in {{mathbb {N}}}), the answer for (*) is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly,

and remark that (c_n) is the root of a polynomial of degree (n-1). We demonstrate that for (n=6,7), (c_n) are algebraic numbers not expressible as radicals over ({{mathbb {Q}}}) (and conjecture this is in fact true for general (n ge 6)).

For Schrödinger operators with potentials that are asymptotically homogeneous of degree (-2), the size of the coupling determines whether it has finite or infinitely many negative eigenvalues. In the latter case, the asymptotic accumulation of these eigenvalues at zero has been determined by Kirsch and Simon. A similar regime occurs for potentials that are not asymptotically monotone but oscillatory. In this case, when the ratio between the amplitude and frequency of oscillation is asymptotically homogeneous of degree (-1), the coupling determines the finiteness of the negative spectrum. We present a new proof of this fact by making use of a ground-state representation. As a consequence of this approach, we derive an asymptotic formula analogous to that of Kirsch and Simon.

For gapped periodic systems (insulators), it has been established that the insulator is topologically trivial (i.e., its Chern number is equal to 0) if and only if its Fermi projector admits an orthogonal basis with finite second moment (i.e., all basis elements satisfy (int |varvec{x}|^2 |w(varvec{x})|^2 ,text {d}{varvec{x}} < infty )). In this paper, we extend one direction of this result to non-periodic gapped systems. In particular, we show that the existence of an orthogonal basis with slightly more decay ((int |varvec{x}|^{2+epsilon } |w(varvec{x})|^2 ,text {d}{varvec{x}} < infty ) for any (epsilon > 0)) is a sufficient condition to conclude that the Chern marker, the natural generalization of the Chern number, vanishes.

We investigate the interior of a dynamical black hole as described by the Einstein–Maxwell-charged-Klein–Gordon system of equations with a cosmological constant, under spherical symmetry. In particular, we consider a characteristic initial value problem where, on the outgoing initial hypersurface, interpreted as the event horizon (mathcal {H}^+) of a dynamical black hole, we prescribe: (a) initial data asymptotically approaching a fixed sub-extremal Reissner–Nordström–de Sitter solution and (b) an exponential Price law upper bound for the charged scalar field. After showing local well-posedness for the corresponding first-order system of partial differential equations, we establish the existence of a Cauchy horizon (mathcal{C}mathcal{H}^+) for the evolved spacetime, extending the bootstrap methods used in the case (Lambda = 0) by Van de Moortel (Commun Math Phys 360:103–168, 2018. https://doi.org/10.1007/s00220-017-3079-3). In this context, we show the existence of (C^0) spacetime extensions beyond (mathcal{C}mathcal{H}^+). Moreover, if the scalar field decays at a sufficiently fast rate along (mathcal {H}^+), we show that the renormalized Hawking mass remains bounded for a large set of initial data. With respect to the analogous model concerning an uncharged and massless scalar field, we are able to extend the known range of parameters for which mass inflation is prevented, up to the optimal threshold suggested by the linear analyses by Costa–Franzen (Ann Henri Poincaré 18:3371–3398, 2017. https://doi.org/10.1007/s00023-017-0592-z) and Hintz–Vasy (J Math Phys 58(8):081509, 2017. https://doi.org/10.1063/1.4996575). In this no-mass-inflation scenario, which includes near-extremal solutions, we further prove that the spacetime can be extended across the Cauchy horizon with continuous metric, Christoffel symbols in (L^2_{text {loc}}) and scalar field in (H^1_{text {loc}}). By generalizing the work by Costa–Girão–Natário–Silva (Commun Math Phys 361:289–341, 2018. https://doi.org/10.1007/s00220-018-3122-z) to the case of a charged and massive scalar field, our results reveal a potential failure of the Christodoulou–Chruściel version of the strong cosmic censorship under spherical symmetry.

We give a simple and self-contained construction of the (P(Phi )) Euclidean quantum field theory in the plane and verify the Osterwalder–Schrader axioms: translational and rotational invariance, reflection positivity and regularity. In the intermediate steps of the construction, we study measures on spheres. In order to control the infinite volume limit, we use the parabolic stochastic quantization equation and the energy method. To prove the translational and rotational invariance of the limit measure, we take advantage of the fact that the symmetry groups of the plane and the sphere have the same dimension.

The spherical Sherrington–Kirkpatrick (SSK) model and its bipartite analog both exhibit the phenomenon that their free energy fluctuations are asymptotically Gaussian at high temperature but asymptotically Tracy–Widom at low temperature. This was proved in two papers by Baik and Lee, for all non-critical temperatures. The case of the critical temperature was recently computed for the SSK model in two separate papers, one by Landon and the other by Johnstone, Klochkov, Onatski, Pavlyshyn. In the current paper, we derive the critical temperature result for the bipartite SSK model. In particular, we find that the free energy fluctuations exhibit a transition when the temperature is in a window of size (n^{-1/3}sqrt{log n}) around the critical temperature, the same window as for the SSK model. Within this transitional window, the asymptotic fluctuations of the free energy are the sum of independent Gaussian and Tracy–Widom random variables.

We study the Schrödinger operator describing a two-dimensional quantum particle moving in the presence of ( N geqslant 1) Aharonov–Bohm magnetic fluxes. We classify all the self-adjont realizations of such an operator, providing an explicit characterization of their domains and actions. Moreover, we examine their spectral and scattering properties, proving in particular the existence and completeness of wave operators in relation with the free dynamics.