An estimate for the number of bound states of the Schrödinger operator in two dimensions

For the Schrödinger operator on let be the number of bound states. One obtains the following estimate:

where and ( is the Euler constant). This estimate holds for all potentials for which the previous integral is finite.

     

Molecular forces caused by the confinement of thermal noise

We have investigated spontaneous surface instabilities of very thin polymer films. Film stability and the wavelength of the dominating unstable mode were found to depend sensitively on the media adjacent to the film. Our experimental results cannot be explained by van der Waals interactions alone. To account for the presence of an additional destabilizing force, we propose that the geometrical confinement of thermally excited acoustic waves gives rise to a force that is strong enough to destabilize thin films. This thermoacoustic effect is of similar magnitude as van der Waals forces.     

On 2-almost contact tachibana manifolds

LeM be a (2m+2)-dimensional Riemannian manifold with two structure vector fieldsξ _r (r=2m+1, 2m+2) and letη ^r =ξ _r ^b be their corresponding covectors (or Pfaffians). These vector fields define onM a 2-almost contact structure. If the 2-formϕ=η ^2m+1∧η ^2m+2 is harmonic, then, following S. Tachibana [12],M is a Tachibana manifold and in this caseM is covered with 2 families of minimal submanifolds tangent toD ^⊥={ξ r} and its complementary orthogonal distributionD ^⊤. On such a manifold a canonical eigenfunction α of the Laplacian is associated. Since the corresponding eingenvalue is negative,M cannot be compact. Any horizontal vector fieldX orthogonal to α^# is a skew-symmetric Killing vector field (see [6]). Next, we assume that the Tachibana manifoldM under consideration is endowed with a framedf-structure defined by an endomorphism ϕ of the tangent bundleTM. Infinitesimal automorphisms of the symplectic form Ω _ϕ are obtained.     

On a class of Einstein hypersurfaces immersed in a Riemannian manifold

Let x:M→ be an isometric immersion of a hypersurface M into an (n+1)-dimensional Riemannian manifold and let ρ _i  (i∈{1,...,n}) be the principal curvatures of M. We denote by E and P the distinguished vector field and the curvature vector field of M, respectively, in the sense of [8].?If M is structured by a P-parallel connection [7], then it is Einsteinian. In this case, all the curvature 2-forms are exact and other properties induced by E and P are stated.?The principal curvatures ρ _i are isoparametric functions and the set (ρ₁,...,ρ _n ) defines an isoparametric system [10].?In the last section, we assume that, in addition, M is endowed with an almost symplectic structure. Then, the dual 1-form π=P ^♭ of P is symplectic harmonic. If M is compact, then its 2nd Betti number b ₂≥1. Received: April 7, 1999; in final form: January 7, 2000?Published online: May 10, 2001     

Analysis of optimality in singular control

The non-negativity, respectively, negativity conditions of the second variation of the performance index, give the sufficient optimality and non-optimality criteria for the singular control. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fast contraction mapping for solving multibody systems

The simulation of complex multibody systems with contacts and friction requires a fast and robust solver for complementarity problems. This work presents an efficient method which can solve large cone-complementarity problems by means of a fixed point iteration. Our method performs like a contractive mapping, providing a monotonic approximation to the exact solution. The algorithm features high computational performance even if thousands of unilateral constraints are added to the system. Also, this scheme fits well in a real-time simulation context because it can be terminated prematurely. As a benchmark for testing the method in challenging situations, we propose the simulation of the granular flow in a fourth-generation bebble-bed nuclear reactor, including 150'000 rigid bodies and more than a million of constraint multipliers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal singular control for systems involving parameters

What we shall take into consideration is represented by the problems of singular optimal control when the index of performance, the differential constraints and the final conditions contain parameters. What is determined is the trajectory of the neighboring extremal for the initial point perturbed at the constraints of the perturbed final manifold. The present study evaluates the variation of the command on the extremal. This one allows obtaining the second variation in the singular case. The sufficient conditions of minimum are a result of imposing the non-negativity of the second variation. The application made analysis the optimality of the singular control for a system in the class of affine differential systems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré’s program of studying the spectrum of the boundary double layer potential is developed in complete generality on closed Lipschitz hypersurfaces in euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2 dimensions. As an application, in the case of planar curves with corners, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory.