We consider the self-adjoint Smilansky Hamiltonian H in L2 R2) associated with the formal differential expression -x2-1/2(y2+y2)-2y(x) in the sub-critical regime, (0, 1). We demonstrate the existence of resonances for H on a countable subfamily of sheets of the underlying Riemann surface whose distance from the physical sheet is finite. On such sheets, we find resonance free regions and characterize resonances for small > 0. In addition, we refine the previously known results on the bound states of H in the weak coupling regime (0+). In the proofs we use Birman-Schwinger principle for H, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.

We show that a compact operator A is a multiple of a positive semi-definite operator if and only if (A B) W (A) W (B), for all (rank one) operators B. An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators A so that equivalence conditions hold.

Shape optimization aims to optimize an objective function by changing the shape of the computational domain. In recent years, shape optimization has received considerable attentions. On the theoretical side there are several publications dealing with the existence of solution and the sensitivity analysis of the problem; see e.g., [6] and references therein.

Let <i>H</i> denote a spherical subgroup within a semisimple algebraic group <i>G</i>. In this paper we study the closures of the finitely many <i>H</i>-orbits in the flag variety of <i>G</i>. Using the language of Frobenius splitting we provide a criterion for these closures to have nice geometric and cohomological properties. We then show how the criterion applies to the spherical subgroups of minimal rank studied by N. Ressayre. Finally, we also provide applications of the criterion to orbit closures which are not multiplicity-free in the sense defined by M. Brion.

We prove the nonlinear asymptotic stability of the gravitational hydrostatic equilibrium for the general equation of state of pressure-density relation in the framework of vacuum free boundary problem of spherically symmetric compressible Navier-Stokes-Poisson equations in three dimensions.The results apply to white dwarfs and polytropes with γ≥2, for which even the linearized asymptotic stability results are not available in literature, to the best of the authors' knowledge. Detailed decay rates of perturbations are given.