We study singular Schrdinger operators with an attractive interaction supported by a closed smooth surface AR3 and analyze their behavior in the vicinity of the critical situation where such an operator has empty discrete spectrum and a threshold resonance. In particular, we show that if A is a sphere and the critical coupling is constant over it, any sufficiently small smooth area-preserving radial deformation gives rise to isolated eigenvalues. On the other hand, the discrete spectrum may be empty for general deformations. We also derive a related inequality for capacities associated with such surfaces.

The problem of maximizing the L p norms of chords connecting points on a closed curve separated by arc length u arises in electrostatic and quantum-mechanical problems. It is known that among all closed curves of fixed length, the unique maximizing shape is the circle for 1 p 2, but this is not the case for sufficiently large values of p. Here we determine the critical value p c (u) of p above which the circle is not a local maximizer finding, in particular, that p c (1 2 L)= 5 2. This corrects a claim made in [P. Exner, EM Harrell, M. Loss, Lett. Math. Phys. 75 (2006) 225].

In this paper, we will develop a new staggered hybridization technique for discontinuous Galerkin methods to discretize linear elastodynamic equations. The idea of hybridization is used extensively in many discontinuous Galerkin methods, but the idea of staggered hybridization is new. Our new approach offers several advantages, namely energy conservation, high-order optimal convergence, preservation of symmetry for the stress tensor, block diagonal mass matrices as well as low dispersion error. The key idea is to use two staggered hybrid variables to enforce the continuity of the velocity and the continuity of the normal component of the stress tensor on a staggered mesh. We prove the stability and the convergence of the proposed scheme in both the semi-discrete and the fully-discrete settings. Numerical results confirm the optimal rate of convergence and show that the method has a superconvergent property for dispersion. Furthermore, an application of this method to Rayleigh wave propagation is presented.

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We consider a pair of isoperimetric problems arising in physics. The first concerns a Schrdinger operator in L^2(\mathbb{R}^2) with an attractive interaction supported on a closed curve , formally given by (<i>x</i>); we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread in L^2(\mathbb{R}^2), homogeneously charged but not conducting, and we ask about the (renormalized) potential-energy minimizer. Both problems reduce to purely geometric questions about inequalities for mean values of chords of . We prove an isoperimetric theorem for <i>p</i>-means of chords of curves when <i>p</i> 2, which implies in particular that the global extrema for the physical problems are always attained when is a circle. The letter concludes with a discussion of the <i>p</i>-means of chords when <i>p</i> &gt; 2.