Commutator relations are used to investigate the spectra of Schrödinger Hamiltonians, H = −∆+ V (x) , acting on functions of a smooth, compact d-dimensional manifold M immersed in R , ν ≥ d+ 1. Here ∆ denotes the Laplace-Beltrami operator, and the real-valued potential–energy function V (x) acts by multiplication. The manifold M may be complete or it may have a boundary, in which case Dirichlet...

Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and diameter, and almost non-negative scalar curvature. Let r = 1 if n = 2, 3 and r = 2 + 1 if n ≥ 4. We show that if the square of the Dirac operator on such a manifold has r small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is ...

We study the question of existence of a Riemannian metric of positive scalar curvature metric on manifolds with the Sullivan–Baas singularities. The manifolds we consider are Spin and simply connected. We prove an analogue of the Gromov–Lawson Conjecture for such manifolds in the case of particular type of singularities. We give an affirmative answer when such manifolds with singularities accep...

In this paper we study the Green function for a second order noncoercive differential operator L on a connected, simply connected homogeneous manifold of negative curvature. Such a manifold is a solvable Lie group S = NA, a semi-direct product of a nilpotent Lie group N and an abelian group A = R. Moreover, for an H belonging to the Lie algebra a of A, the real parts of the eigenvalues of Adexp...

We consider a singular Schrödinger operator in L(R) written formally as −∆ − βδ(x − γ) where γ is a C smooth open arc in R of length L with regular ends. It is shown that the jth negative eigenvalue of this operator behaves in the strong-coupling limit, β → +∞, asymptotically as Ej(β) = − β 4 + μj +O ( log β β ) , where μj is the jth Dirichlet eigenvalue of the operator − d 2 ds2 − κ(s) 2 4 on ...

Let Γ be a discrete group, and let M be a closed spin manifold of dimension m > 3 with π1(M) = Γ. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L-rho invariant ρ(2) and the delocalized eta invariant η associated to the Dirac operator on M in order to get information about the space of metrics with positive scalar curvature. In particular ...

Let Γ be a discrete group, and let M be a closed spin manifold of dimension m > 3 with π1(M) = Γ. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L-rho invariant ρ(2) and the delocalized eta invariant η associated to the Dirac operator on M in order to get information about the space of metrics with positive scalar curvature. In particular ...

Let Γ be a discrete group, and let M be a closed spin manifold of dimension m > 3 with π1(M) = Γ. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L-rho invariant ρ(2) and the delocalized eta invariant η associated to the Dirac operator on M in order to get information about the space of metrics with positive scalar curvature. In particular ...

Let Γ be a discrete group, and let M be a closed spin manifold of dimension m > 3 with π1(M) = Γ. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L-rho invariant ρ(2) and the delocalized eta invariant η associated to the Dirac operator on M in order to get information about the space of metrics with positive scalar curvature. In particular ...

‎let $m^n$ be an $n(ngeq 3)$-dimensional complete connected and‎ ‎oriented spacelike hypersurface in a de sitter space or an anti-de‎ ‎sitter space‎, ‎$s$ and $k$ be the squared norm of the second‎ ‎fundamental form and gauss-kronecker curvature of $m^n$‎. ‎if $s$ or‎ ‎$k$ is constant‎, ‎nonzero and $m^n$ has two distinct principal‎ ‎curvatures one of which is simple‎, ‎we obtain some‎ ‎charact...