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.

No abstract uploaded!

No abstract uploaded!

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.

Various qualitative methods for analyzing the discrete spectrum of SchrOdinger operators have been elaborated during the last three decades-for a review see Eef. 1, Section XIII. 3. They are of a great physical interest, especially because they can provide us with an information about -)*the few-body and many-body Hamlltonians, for which the quantitative methods are usually difficult to be applied. lt is not strange, therefore, that new re3ulta of this type continue to appear.