We shall consider the following Dirichlet eigenvalue problem on a smooth bounded domain S~ eRn, I where V is a nonnegative function defined on,~. As is well-known, the eigenvalues of problem (1.1) can be interpreted as the energy levels of a particle travelling under an external force field of a potential q in Rn, where and the corresponding eigenfunctions are wave functions of the Schrodinger equation-J~+ qu= lu. Furthermore, the set of eigenvalues {A,} of (1.1) are nonnegative and can be arranged in a nondecreasing order as follows, It is a significant problem to find a lower bound for~, 1 in terms of the geometry of Q. This subject has been studied extensively by many authors. A rather precise bound in the case V== 0 was worked out not only for a bounded domain in but actually valid for a general Riemannian manifold with certain curvature conditions; we refer to [4] for these recent develop-ments. Nevertheless, very little is known about the obvious interesting question of how big the gap is between 2 and l. There are both physical

In this short survey paper, we shall discuss certain recent results in classical gravity. Our main attention will be restricted to two topics in which we have been involved; the positive mass conjecture and its extensions to the case with horizons, including the Penrose conjecture (Part I), and the interaction of gravity with other force fields and quantum-mechanical particles (Part II).

The paper discusses quantum graphs with a vertex coupling which interpolates between the common one of the type and a coupling introduced recently by two of the authors which exhibits a preferred orientation. Describing the interpolation family in terms of circulant matrices, we analyze the spectral and scattering property of such vertices, and investigate the band spectrum of the corresponding square lattice graph.

In the past decade, tremendous efforts have been made towards understanding fermionic symmetry protected topological (FSPT) phases in interacting systems. Nevertheless, for systems with continuum symmetry, e.g., electronic insulators, it is still unclear how to construct an exactly solvable model with a finite dimensional Hilbert space in general. In this paper, we give a lattice model construction and classification for 2D interacting electronic insulators. Based on the physical picture of U(1)_f-charge decorations, we illustrate the key idea by considering the well known 2D interacting topological insulator. Then we generalize our construction to an arbitrary 2D interacting electronic insulator with symmetry Gf=U(1)_f ⋊_{ρ_1,ω_2} G, where U(1)_f is the charge conservation symmetry and ρ_1,ω_2 are additional data which fully characterize the group structure of G_f. Finally we study more examples, including the full interacting classification of 2D crystalline topological insulators.

We analyze spectral properties of a leaky wire model with a potential bias. It describes a two-dimensional quantum particle exposed to a potential consisting of two parts. One is an attractive -interaction supported by a non-straight, piecewise smooth curve L dividing the plane into two regions of which one, the interior, is convex. The other interaction component is a constant positive potential V0 in one of the regions. We show that in the critical case, V0 = 2, the discrete spectrum is non-void if and only if the bias is supported in the interior. We also analyze the non-critical situations, in particular, we show that in the subcritical case, V0 < 2, the system may have any finite number of bound states provided the angle between the asymptotes of L is small enough.