Annales Henri Poincaré最新文献

We are concerned in this paper with the real eigenfunctions of Schrödinger operators. We prove an asymptotic upper bound for the number of their nodal domains, which implies in particular that the inequality stated in Courant’s theorem is strict, except for finitely many eigenvalues. Results of this type originated in 1956 with Pleijel’s theorem on the Dirichlet Laplacian and were obtained for some classes of Schrödinger operators by the first author, alone and in collaboration with B. Helffer and T. Hoffmann-Ostenhof. Using methods in part inspired by work of the second author on Neumann and Robin Laplacians, we greatly extend the scope of these previous results.

The Reissner–Weyl–Nordström (RWN) spacetime of a point nucleus features a naked singularity for the empirically known nuclear charges Ze and masses (M = A(Z,N)m_{textrm{p}}), where (m_{textrm{p}}) is the proton mass and (A(Z,N)approx Z+N) the atomic mass number, with Z the number of protons and N the number of neutrons in the nucleus. The Dirac Hamiltonian for a test electron with mass (m_{textrm{e}}), charge (-,e), and anomalous magnetic moment (mu _a (approx -, frac{1}{4pi }frac{e^3}{m_{textrm{e}}c^2})) in the electrostatic RWN spacetime of such a “naked point nucleus” is known to be essentially self-adjoint, with a spectrum that consists of the union of the essential spectrum ((-,infty ,-,m_{textrm{e}}c^2]cup [m_{textrm{e}}c^2, infty )) and a discrete spectrum of infinitely many eigenvalues in the gap ((-,m_{textrm{e}}c^2,m_{textrm{e}}c^2)), having (m_{textrm{e}}c^2) as accumulation point. In this paper, the discrete spectrum is characterized in detail for the first time, for all (Zle 45) and A that cover all known isotopes. The eigenvalues are mapped one-to-one to those of the traditional Dirac hydrogen spectrum. Numerical evaluations that go beyond (Z=45) into the realm of not-yet-produced hydrogenic ions are presented, too. A list of challenging open problems concludes this publication.

Boundedness is demonstrated for the fifth derivative of the one-particle reduced density matrix for non-relativistic Coulombic wavefunctions in the vicinity of the diagonal. To prove this result, improved pointwise bounds are obtained for cluster derivatives of wavefunctions involving multiple clusters.

The precise renormalizable interactions in the bosonic sector of electroweak theory are intrinsically determined in the autonomous approach to perturbation theory. This proceeds directly on the Hilbert–Fock space built on the Wigner unirreps of the physical particles, with their given masses: those of three massive vector bosons, a photon, and a massive scalar (the “higgs”). Neither “gauge choices” nor an unobservable “mechanism of spontaneous symmetry breaking” is invoked. Instead, to proceed on Hilbert space requires using string-localized fields to describe the vector bosons. In such a framework, the condition of string independence of the ({mathbb {S}})-matrix yields consistency constraints on the coupling coefficients, the essentially unique outcome being the experimentally known one. The analysis can be largely carried out for other configurations of massive and massless vector bosons, paving the way towards consideration of consistent mass patterns beyond those of the electroweak theory.

The Dirac vacuum is a nonlinear polarisable medium rather than an empty space. This nonlinear behaviour starts to be significant for extremely large electromagnetic fields such as the magnetic field on the surface of certain neutron stars. Even though the null temperature case was deeply studied in the past decades, the problem at nonzero temperature needs to be better understood. In this work, we present the first rigorous derivation of the one-loop effective magnetic Lagrangian at positive temperature, a nonlinear functional describing the free energy of the Dirac vacuum in a classical magnetic field. After introducing our model, we properly define the free energy functional using the Pauli–Villars regularisation technique in order to remove the worst ultraviolet divergences, which represent a well-known issue of the theory. The study of the properties of this functional is addressed before focusing on the limit of slowly varying classical magnetic fields. In this regime, we prove the convergence of this functional to the Euler–Heisenberg formula with thermal corrections, recovering the effective Lagrangian first derived by Dittrich (Phys Rev D 19:2385–2390, 1979).

We investigate the negative part of the spectrum of the operator (-partial ^2 - mu ) on (L^2(mathbb {R})), where a locally finite Radon measure (mu ge 0) serves as a potential. We obtain estimates for the eigenvalue counting function, for individual eigenvalues and estimates of the Lieb–Thirring type. A crucial tool for our estimates is Otelbaev’s function, a certain average of the measure-potential (mu ), which is used both in the proofs and the formulation of most of the results.

We study the inverse problem of recovering primordial perturbations from anisotropies of the cosmic microwave background (CMB) using the kinetic model. Mathematically, the problem in concern is the inverse source problem for the linear Boltzmann equation with measurements on some Cauchy surface. We obtain two stable determination results for generic absorption coefficients and scattering kernels.

We study the Dirac Hamiltonian in dimension two with a mass term and a large momentum regularization and show that bulk-edge correspondence fails. Despite a well-defined bulk topological index (the Chern number), the number of edge modes depends on the boundary condition. The origin of this anomaly is rooted in the unbounded nature of the spectrum. It is detected with Levinson’s theorem from scattering theory and quantified via an anomalous winding number at infinite energy, dubbed ghost charge. First, we classify, up to equivalence, all self-adjoint boundary conditions, using Schubert cell decomposition of a Grassmannian. Then, we investigate which ones are anomalous. We expand the scattering amplitude near infinite energy, for which a dominant scale captures the asymptotic winding number. Remarkably, this can be achieved for every self-adjoint boundary condition, leading to an exhaustive anomaly classification. It shows that anomalies are ubiquitous and typical. Boundary conditions with a ghost charge of 2 are also revealed within the process.

We prove that all nice holomorphic vertex operator superalgebras (VOSAs) with central charge at most 24 and with non-trivial odd part are unitary, apart from the hypothetical ones arising as fake copies of the shorter moonshine VOSA or of the latter tensorized with a real free fermion VOSA. Furthermore, excluding the ones with central charge 24 of glueing type III and with no real free fermion, we show that they are all strongly graded-local. In particular, they naturally give rise to holomorphic graded-local conformal nets. In total, we are able to prove that 910 of the 969 nice holomorphic VOSAs with central charge 24 and with non-trivial odd part are strongly graded-local, without counting hypothetical fake copies of the shorter moonshine VOSA tensorized with a real free fermion VOSA.

We propose a geometric characterisation of the topological string partition functions associated with the local Calabi–Yau (CY) manifolds used in the geometric engineering of (d=4), ({mathcal {N}}=2) supersymmetric field theories of class ({mathcal {S}}). A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated with the quantum curves by expansions of generalised theta series type. It turns out that the partition functions are in one-to-one correspondence with preferred coordinates on the moduli spaces of quantum curves defined using the Exact WKB method. The coordinates defined in this way jump across certain loci in the moduli space. The changes of normalization of the tau-functions associated with these jumps define a natural line bundle playing a key role in the geometric characterisation of the topological string partition functions proposed here.