6,856 research outputs found

    Avoided intersections of nodal lines

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    We consider real eigen-functions of the Schr\"odinger operator in 2-d. The nodal lines of separable systems form a regular grid, and the number of nodal crossings equals the number of nodal domains. In contrast, for wave functions of non integrable systems nodal intersections are rare, and for random waves, the expected number of intersections in any finite area vanishes. However, nodal lines display characteristic avoided crossings which we study in the present work. We define a measure for the avoidance range and compute its distribution for the random waves ensemble. We show that the avoidance range distribution of wave functions of chaotic systems follow the expected random wave distributions, whereas for wave functions of classically integrable but quantum non-separable wave functions, the distribution is quite different. Thus, the study of the avoidance distribution provides more support to the conjecture that nodal structures of chaotic systems are reproduced by the predictions of the random waves ensemble.Comment: 12 pages, 4 figure

    The radial curvature of an end that makes eigenvalues vanish in the essential spectrum II

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    Under the quadratic-decay-conditions of the radial curvatures of an end, we shall derive growth estimates of solutions to the eigenvalue equation and show the absence of eigenvalues.Comment: " \ge " in the conditions (4)(*_4) and (5)(*_5) should be replaced by ">>". γn12(ba)\gamma \ge \frac{n-1}{2}(b-a) in the conclusion of Theorem 1.3 should be replaced by γ>n12(ba)\gamma > \frac{n-1}{2}(b-a); trivial miss-calculatio

    On the magnitude of spheres, surfaces and other homogeneous spaces

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    In this paper we define the magnitude of metric spaces using measures rather than finite subsets as had been done previously and show that this agrees with earlier work with Leinster in arXiv:0908.1582. An explicit formula for the magnitude of an n-sphere with its intrinsic metric is given. For an arbitrary homogeneous Riemannian manifold the leading terms of the asymptotic expansion of the magnitude are calculated and expressed in terms of the volume and total scalar curvature of the manifold. In the particular case of a homogeneous surface the form of the asymptotics can be given exactly up to vanishing terms and this involves just the area and Euler characteristic in the way conjectured for subsets of Euclidean space in previous work.Comment: 21 pages. Main change from v1: details added to proof of Theorem

    A Spectral Bernstein Theorem

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    We study the spectrum of the Laplace operator of a complete minimal properly immersed hypersurface MM in Rn+1\R^{n+1}. (1) Under a volume growth condition on extrinsic balls and a condition on the unit normal at infinity, we prove that MM has only essential spectrum consisting of the half line [0,+)[0, +\infty). This is the case when limr~+r~κi=0\lim_{\tilde{r}\to +\infty}\tilde{r}\kappa_i=0, where r~\tilde{r} is the extrinsic distance to a point of MM and κi\kappa_i are the principal curvatures. (2) If the κi\kappa_i satisfy the decay conditions κi1/r~|\kappa_i|\leq 1/\tilde{r}, and strict inequality is achieved at some point yMy\in M, then there are no eigenvalues. We apply these results to minimal graphic and multigraphic hypersurfaces.Comment: 16 pages. v2. Final version: minor revisions, we add Proposition 3.2. Accepted for publication in the Annali di Matematica Pura ed Applicata, on the 05/03/201

    Local Asymmetry and the Inner Radius of Nodal Domains

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    Let M be a closed Riemannian manifold of dimension n. Let f be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue \lambda. We show that the volume of {f>0} inside any ball B whose center lies on {f=0} is > C|B|/\lambda^n. We apply this result to prove that each nodal domain contains a ball of radius > C/\lambda^n.Comment: 12 pages, 1 figure; minor corrections; to appear in Comm. PDE

    Lower bounds for nodal sets of Dirichlet and Neumann eigenfunctions

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    Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {\phi=0}.Comment: 7 page

    Advanced bladder technology. Gas impermeable protein films and laminates Final report

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    Gas impermeable and cryogenic flexible protein film

    Multiple Breathers on a Vortex Filament

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    In this paper we investigate the correspondence between the Da Rios-Betchov equation, which appears in the three-dimensional motion of a vortex filament, and the nonlinear Schrödinger equation. Using this correspondence we map a set of solutions corresponding to breathers in the nonlinear Schrödinger equation to waves propagating along a vortex filament. The work presented generalizes the recently derived family of vortex configurations associated with these breather solutions to a wider class of configurations that are associated with combination homoclinic/heteroclinic orbits of the 1D self-focussing nonlinear Schrödinger equation. We show that by considering these solutions of the governing nonlinear Schrödinger equation, highly nontrivial vortex filament configurations can be obtained that are associated with a pair of breather excitations. These configurations can lead to loop-like excitations emerging from an otherwise weakly perturbed helical vortex. The results presented further demonstrate the rich class of solutions that are supported by the Da Rios-Betchov equation that is recovered within the local induction approximation for the motion of a vortex filament
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