On the semiclassical approximation to the eigenvalue gap of Schrödinger operators

We consider two types of Schrödinger operators $H(t)=?d^2/d{x}^2+q(x)+t\cos x$ and $H(t)=?d^2/d{x}^2+q(x)+A\cos (tx)$ defined on $L^2(\mathbb{R})$, where q is an even potential that is bounded from below, A is a constant, and $t>0$ is a parameter. We assume that $H(t)$ has at least two eigenvalues below its essential spectrum; and we denote by ${\lambda }_1(t)$ and ${\lambda }_2(t)$ the lowest eigenvalue and the second one, respectively. The purpose of this paper is to study the asymptotics of the gap $\Gamma (t)={\lambda }_2(t)?{\lambda }_1(t)$ in the limit as $t\to \infty$.     

A mapping connected with the Schur–Szegő composition

Every monic polynomial in one variable of the form $(x+1)S$, $\mathrm{deg}S=n?1$, is presentable in a unique way as a Schur–Szeg? composition of $n?1$ polynomials of the form ${(x+1)}^{n?1}(x+a_i)$. We prove geometric properties of the affine mapping associating to the coefficients of S the $(n?1)$-tuple of values of the elementary symmetric functions of the numbers $a_i$. To cite this article: V.P. Kostov, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On triviality of the Kashiwara–Vergne problem for quadratic Lie algebras

We show that the Kashiwara–Vergne (KV) problem for quadratic Lie algebras (that is, Lie algebras admitting an invariant scalar product) reduces to the problem of representing the Campbell–Hausdorff series in the form $\ln (e^x{e}^y)=x+y+[x,a(x,y)]+[y,b(x,y)]$, where $a(x,y)$ and $b(x,y)$ are Lie series in x and y. This observation explains the existence of explicit rational solutions of the quadratic KV problem, whereas constructing an explicit rational solution of the full KV problem would probably require the knowledge of a rational Drinfeld associator. It also gives, in the case of quadratic Lie algebras, a direct proof of the Duflo theorem (implied by the KV problem). To cite this article: A. Alekseev, C. Torossian, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): $\dot{x}=y?12x^2+16y^2$, $\dot{y}=?x?16xy$, and (r20): $\dot{x}=y+4x^2$, $\dot{y}=?x+16xy$, and the periodic orbits of the quadratic isochronous centers $(S_1):\dot{x}=?y+x^2?y^2$, $\dot{y}=x+2xy$, and $(S_2):\dot{x}=?y+x^2$, $\dot{y}=x+xy$. The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system $(S_1)$ and $(S_2)$ are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line $y=0$. It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively $4n?3(n\ge 4)$ and $4n+3(n\ge 3)$ counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers $(S_1)$ and $(S_2)$ are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively.     

Zeros of Jacobi functions of second kind

The number of zeros in $(-1,1)$ of the Jacobi function of second kind $Q_n^{(\alpha ,\beta )}(x)$, $\alpha ,\beta >-1$, i.e. the second solution of the differential equation$(1-x^2)y^{″}(x)+(\beta -\alpha -(\alpha +\beta +2)x)y^{\prime }(x)+n(n+\alpha +\beta +1)y(x)=0\text{,}$is determined for every $n\in \mathbb{N}$ and for all values of the parameters $\alpha >-1$ and $\beta >-1$. It turns out that this number depends essentially on $\alpha$ and $\beta$ as well as on the specific normalization of the function $Q_n^{(\alpha ,\beta )}(x)$. Interlacing properties of the zeros are also obtained. As a consequence of the main result, we determine the number of zeros of Laguerre's and Hermite's functions of second kind.     

On a nonlinear Schrödinger equation with a localizing effect

We consider the nonlinear Schrödinger equation associated to a singular potential of the form $a{|u|}^{?(1?m)}u+bu$, for some $m\in (0,1)$, on a possible unbounded domain. We use some suitable energy methods to prove that if $\mathrm{Re}(a)+\mathrm{Im}(a)>0$ and if the initial and right hand side data have compact support then any possible solution must also have a compact support for any $t>0$. This property contrasts with the behavior of solutions associated to regular potentials $(m?1)$. Related results are proved also for the associated stationary problem and for self-similar solution on the whole space and potential $a{|u|}^{?(1?m)}u$. The existence of solutions is obtained by some compactness methods under additional conditions. To cite this article: P. Bégout, J.I. Díaz, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Paramétrisation des points algébriques de degré donné sur la courbe d'équation affine y3=x(x−1)(x−2)(x−3)

We give a parameterization of the algebraic points of given degree over $\mathbb{Q}$ on the curve$y^3=x(x?1)(x?2)(x?3)$ This result extends a previous result of E.F. Schaefer who described in Schaefer (1998) [1] the set of algebraic points of degree ?3 over $\mathbb{Q}$.