Lower bounds for the spinless Salpeter equation

We obtain lower bounds on the ground state energy, in one and three dimensions, for the spinless Salpeter equation (Schr\"odinger equation with a relativistic kinetic energy operator) applicable to potentials for which the attractive parts are in $L^p(\R^n)$ for some $p>n$ ($n=1$ or 3). An extension to confining potentials, which are not in $L^p(\R^n)$, is also presented.Comment: 11 pages, 2 figures. Contribution to a special issue of Journal of Nonlinear Mathematical Physics in honour of Francesco Calogero on the occasion of his seventieth birthda

Upper limit on the number of bound states of the spinless Salpeter equation

We obtain, using the Birman-Schwinger method, upper limits on the total number of bound states and on the number of $\ell$-wave bound states of the semirelativistic spinless Salpeter equation. We also obtain a simple condition, in the ultrarelativistic case ($m=0$), for the existence of at least one $\ell$-wave bound states: $C(\ell,p/(p-1))$ $\int_0^{\infty}dr r^{p-1} |V^-(r)|^p\ge 1$, where $C(\ell,p/(p-1))$ is a known function of $\ell$ and $p>1$.Comment: 18 page

Upper limit on the critical strength of central potentials in relativistic quantum mechanics

In the context of relativistic quantum mechanics, where the Schr\"odinger equation is replaced by the spinless Salpeter equation, we show how to construct a large class of upper limits on the critical value, $g_{\rm{c}}^{(\ell)}$, of the coupling constant, $g$, of the central potential, $V(r)=-g v(r)$. This critical value is the value of $g$ for which a first $\ell$-wave bound state appears.Comment: 8 page

Lower limit in semiclassical form for the number of bound states in a central potential

We identify a class of potentials for which the semiclassical estimate $N^{\text{(semi)}}=\frac{1}{\pi}\int_0^\infty dr\sqrt{-V(r)\theta[-V(r)]}$ of the number $N$ of (S-wave) bound states provides a (rigorous) lower limit: $N\ge {{N^{\text{(semi)}}}}$, where the double braces denote the integer part. Higher partial waves can be included via the standard replacement of the potential $V(r)$ with the effective $\ell$-wave potential $V_\ell^{\text{(eff)}}(r)=V(r)+\frac{\ell(\ell+1)}{r^2}$. An analogous upper limit is also provided for a different class of potentials, which is however quite severely restricted.Comment: 9 page

Wrinkles and folds in a fluid-supported sheet of finite size

A laterally confined thin elastic sheet lying on a liquid substrate displays regular undulations, called wrinkles, characterized by a spatially extended energy distribution and a well-defined wavelength $\lambda$. As the confinement increases, the deformation energy is progressively localized into a single narrow fold. An exact solution for the deformation of an infinite sheet was previously found, indicating that wrinkles in an infinite sheet are unstable against localization for arbitrarily small confinement. We present an extension of the theory to sheets of finite length $L$, accounting for the experimentally observed wrinkle-to-fold transition. We derive an exact solution for the periodic deformation in the wrinkled state, and an approximate solution for the localized, folded state. We show that a second-order transition between these two states occurs at a critical confinement $\Delta_F=\lambda^2/L$.Comment: 15 page

Existence of mesons after deconfinement

We investigate the possibility for a quark-antiquark pair to form a bound state at temperatures higher than the critical one ($T>T_c$), thus after deconfinement. Our main goal is to find analytical criteria constraining the existence of such mesons. Our formalism relies on a Schr\"{o}dinger equation for which we study the physical consequences of both using the free energy and the internal energy as potential term, assuming a widely accepted temperature-dependent Yukawa form for the free energy and a recently proposed nonperturbative form for the screening mass. We show that using the free energy only allows for the 1S bottomonium to be bound above $T_c$, with a dissociation temperature around $1.5\times T_c$. The situation is very different with the internal energy, where we show that no bound states at all can exist in the deconfined phase. But, in this last case, quasi-bound states could be present at higher temperatures because of a positive barrier appearing in the potential.Comment: 14 pages, 3 figures; only the case T>T_c is discussed in v

Construction and test of a moving boundary model for negative streamer discharges

Starting from the minimal model for the electrically interacting particle densities in negative streamer discharges, we derive a moving boundary approximation for the ionization fronts. The boundary condition on the moving front is found to be of 'kinetic undercooling' type. The boundary approximation, whose first results have been published in [Meulenbroek et al., PRL 95, 195004 (2005)], is then tested against 2-dimensional simulations of the density model. The results suggest that our moving boundary approximation adequately represents the essential dynamics of negative streamer fronts.Comment: 10 pages, 7 figures; submitted to Phys. Rev.

From cylindrical to stretching ridges and wrinkles in twisted ribbons

Twisted ribbons subjected to a tension exhibit a remarkably rich morphology, from smooth and wrinkled helicoids, to cylindrical or faceted patterns. These shapes are intimately related to the instability of the natural, helicoidal symmetry of the system, which generates both longitudinal and transverse stresses, thereby leading to buckling of the ribbon. In this paper, we focus on the tessellation patterns made of triangular facets. Our experimental observations are described within an "asymptotic isometry" approach that brings together geometry and elasticity. The geometry consists of parametrized families of surfaces, isometric to the undeformed ribbon in the singular limit of vanishing thickness and tensile load. The energy, whose minimization selects the favored structure among those families, is governed by the tensile work and bending cost of the pattern. This framework describes the coexistence lines in a morphological phase diagram, and determines the domain of existence of faceted structures.Comment: 5 pages, 4 figures; Supplemental material: 4 page

On the decrease of the number of bound states with the increase of the angular momentum

For the class of central potentials possessing a finite number of bound states and for which the second derivative of $r V(r)$ is negative, we prove, using the supersymmetric quantum mechanics formalism, that an increase of the angular momentum $\ell$ by one unit yields a decrease of the number of bound states of at least one unit: $N_{\ell+1}\le N_{\ell}-1$. This property is used to obtain, for this class of potential, an upper limit on the total number of bound states which significantly improves previously known results

How Geometry Controls the Tearing of Adhesive Thin Films on Curved Surfaces

Flaps can be detached from a thin film glued on a solid substrate by tearing and peeling. For flat substrates, it has been shown that these flaps spontaneously narrow and collapse in pointy triangular shapes. Here we show that various shapes, triangular, elliptic, acuminate or spatulate, can be observed for the tears by adjusting the curvature of the substrate. From combined experiments and theoretical models, we show that the flap morphology is governed by simple geometric rules.Comment: 6 pages, 5 figure