Annales Henri Poincaré最新文献

We obtain Hölder stability estimates for the inverse Steklov and Calderón problems for Schrödinger operators corresponding to a special class of (L^2) radial potentials on the unit ball. These results provide an improvement on earlier logarithmic stability estimates obtained in Daudé et al. (J Geom Anal 31(2):1821–1854, 2021) in the case of the Schrödinger operators related to deformations of the closed Euclidean unit ball. The main tools involve: (i) A formula relating the difference of the Steklov spectra of the Schrödinger operators associated to the original and perturbed potential to the Laplace transform of the difference of the corresponding amplitude functions introduced by Simon (Ann Math 150:1029–1057, 1999) in his representation formula for the Weyl-Titchmarsh function, and (ii) a key moment stability estimate due to Still (J Approx Theory 45:26–54, 1985). It is noteworthy that with respect to the original Schrödinger operator, the type of perturbation being considered for the amplitude function amounts to the introduction of a finite number of negative eigenvalues and of a countable set of negative resonances which are quantified explicitly in terms of the eigenvalues of the Laplace-Beltrami operator on the boundary sphere.

We consider the Cauchy problem of one-dimensional dissipative nonlinear Schrödinger equations with a critical power nonlinearity. In the previous work, Ogawa–Sato (Nonlinear Differ Equ Appl 27:18, 2020) showed the upper (L^2)-decay estimate of dissipative solutions in the analytic class. In this paper, we show that (L^2)-decay rate obtained in the previous work is optimal for special solutions by obtaining the lower (L^2)-decay estimate.

The degeneracies of 1/4 BPS states with unit torsion in heterotic string theory compactified on a six torus are given in terms of the Fourier coefficients of the reciprocal of the Igusa cusp Siegel modular form (Phi _{10}) of weight 10. We use the symplectic symmetries of the latter to construct a fine-grained Rademacher-type expansion which expresses these BPS degeneracies as a regularized sum over residues of the poles of (1/Phi _{10}). The construction uses two distinct (textrm{SL}(2, {mathbb {Z}})) subgroups of (textrm{Sp}(2, {mathbb {Z}})) which encode multiplier systems, Kloosterman sums and Eichler integrals appearing therein. Additionally, it shows how the polar data are explicitly built from the Fourier coefficients of (1/eta ^{24}) by means of a continued fraction structure.

We consider one-dimensional Schrödinger operators with generalized almost periodic potentials with jump discontinuities and (delta )-interactions. For operators of this kind, we introduce a rotation number in the spirit of Johnson and Moser. To do this, we introduce the concept of almost periodicity at a rather general level, and then the almost periodic function with jump discontinuities and (delta )-interactions as an application.

We consider the Laplacian on a metric graph, equipped with Robin ((delta )-type) vertex condition at some of the graph vertices and Neumann–Kirchhoff condition at all others. The corresponding eigenvalues are called Robin eigenvalues, whereas they are called Neumann eigenvalues if the Neumann–Kirchhoff condition is imposed at all vertices. The sequence of differences between these pairs of eigenvalues is called the Robin–Neumann gap. We prove that the limiting mean value of this sequence exists and equals a geometric quantity, analogous to the one obtained for planar domains by Rudnick et al. (Commun Math Phys, 2021. arXiv:2008.07400). Moreover, we show that the sequence is uniformly bounded and provide explicit upper and lower bounds. We also study the possible accumulation points of the sequence and relate those to the associated probability distribution of the gaps. To prove our main results, we prove a local Weyl law, as well as explicit expressions for the second moments of the eigenfunction scattering amplitudes.

We consider an abstract sequence ({A_n}_{n=1}^infty ) of closed symmetric operators on a separable Hilbert space ({mathcal {H}}). It is assumed that all (A_n)’s have equal deficiency indices (k, k) and thus self-adjoint extensions ({B_n}_{n=1}^infty ) exist and are parametrized by partial isometries ({U_n}_{n=1}^infty ) on ({mathcal {H}}) according to von Neumann’s extension theory. Under two different convergence assumptions on the (A_n)’s we give the precise connection between strong resolvent convergence of the (B_n)’s and strong convergence of the (U_n)’s.

Exceptional orthogonal Hermite polynomials have been linked to the k-step extension of the harmonic oscillator. The exceptional polynomials allow the existence of different supercharges in the Darboux–Crum and Krein–Adler constructions. They also allow the existence of different types of ladder relations and their associated recurrence relations. The existence of such relations is a unique property of these polynomials. Those relations have been used to construct 2D models which are superintegrable and display an interesting spectrum, degeneracies and finite-dimensional unitary representations. In previous works, only the physical or polynomial part of the spectrum was discussed. It is known that the general solutions are associated with other types of recurrence/ladder relations. We discuss in detail the case of the exceptional Hermite polynomials (X_2^{(1)}) and explicitly present new chains obtained by acting with different types of ladder operators. We exploit a recent result by one of the authors (Chalifour and Grundland in Ann Henri Poincaré 21:3341, 2020), where the general analytic solution was constructed and connected with the non-degenerate confluent Heun equation. The analogue Rodrigues formulas for the general solution are constructed. The set of finite states from which the other states can be obtained algebraically is not unique, but the vanishing arrow and diagonal arrow from the diagram of the 2-chain representations can be used to obtain minimal sets. These Rodrigues formulas are then exploited, not only to construct the states, polynomial and non-polynomial, in a purely algebraic way, but also to obtain coefficients from the action of ladder operators in an algebraic manner based on further commutation relations between monomials in the generators.

The C*-algebraic construction of QFT by Buchholz and one of us relies on the causal structure of space-time and a classical Lagrangian. In one of our previous papers, we have introduced additional structure into this construction, namely an action of symmetries, which is related to fixing renormalization conditions. This action characterizes anomalies and satisfies a cocycle condition which is summarized in the unitary anomalous Master Ward identity. Here (using perturbation theory) we show how this cocycle condition is related to the Wess–Zumino consistency relation and the consistency relation for the anomaly in the BV formalism, where the latter follows from the generalized Jacobi identity for the associated (L_infty )-algebra. In addition, we give a proof that perturbative agreement (i.e., independence of a perturbative QFT on the splitting of the Lagrangian into free and interacting parts) can be achieved by finite renormalizations.

We consider the ((1+3))-dimensional Einstein equations with negative cosmological constant coupled to a spherically symmetric, massless scalar field and study perturbations around the anti-de Sitter spacetime. We derive the resonant systems, pick out vanishing secular terms and discuss issues related to small divisors. Most importantly, we rigorously establish (sharp, in most of the cases) asymptotic behaviour for all the interaction coefficients. The latter is based on uniform estimates for the eigenfunctions associated to the linearized operator as well as on some oscillatory integrals.

A class of isotropic and scale-invariant strain energy functions is given for which the corresponding spherically symmetric elastic motion includes bodies whose diameter becomes infinite with time or collapses to zero in finite time, depending on the sign of the residual pressure. The bodies are surrounded by vacuum so that the boundary surface forces vanish, while the density remains strictly positive. The body is subject only to internal elastic stress.