Annales Henri Poincaré最新文献

We prove an upper bound on the energy density of the dilute spin-(frac{1}{2}) Fermi gas capturing the leading correction to the kinetic energy (8pi a rho _uparrow rho _downarrow ) with an error of size smaller than (arho ^{2}(a^3rho )^{1/3-varepsilon }) for any (varepsilon > 0), where a denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin et al. (Nucl Phys A 176(2):237–260, 1971. https://doi.org/10.1016/0375-9474(71)90267-3).

We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order ( N^{kappa -1}), for some (kappa >0). For the range ( kappa in [0, frac{1}{43})), Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for ( kappa in [0, frac{1}{20})) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).

In this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil–Petersson surfaces of large genus g with (o(sqrt{g})) cusps, we prove convergence of the spectral density to the spectral density of the hyperbolic plane, with quantitative error estimates. This result implies upper bounds on spectral counting functions and multiplicities, as well as a uniform Weyl law, true for typical hyperbolic surfaces equipped with any nontrivial spin structure.

In general relativity, time functions are crucial objects whose existence and properties are intimately tied to the causal structure of a spacetime and also to the initial value formulation of the Einstein equations. In this work we establish all fundamental classical existence results on time functions in the setting of Lorentzian (pre-)length spaces (including causally plain continuous spacetimes, closed cone fields and even more singular spaces). More precisely, we characterize the existence of time functions by K-causality, show that a modified notion of Geroch’s volume functions are time functions if and only if the space is causally continuous, and lastly, characterize global hyperbolicity by the existence of Cauchy time functions, and Cauchy sets. Our results thus inevitably show that no manifold structure is needed in order to obtain suitable time functions.

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.