Gradient estimate for Schrödinger operators on manifolds

In this paper, we prove a new localized version of a gradient estimate for Schrödinger operators on the complete manifolds without boundary and with Ricci curvature bounded below by a negative constant. As its application, we derive the Liouville type theorem, the Harnack inequality and the Gaussian lower bound of the heat kernel of Schrödinger operators.     

The Egorov theorem for transverse Dirac-type operators on foliated manifolds

Egorov’s theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first-order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel transport by the parallel transport determined by the transverse Levi-Civita connection. These results allow us to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.     

A separation property for magnetic Schrödinger operators on Riemannian manifolds

We consider a Schrödinger differential expression L=_ΔA+q$L={\Delta }_A+q$ on a complete Riemannian manifold (M,g)$(M,g)$ with metric g$g$, where _ΔA${\Delta }_A$ is the magnetic Laplacian on M$M$ and q≥0$q\ge 0$ is a locally square integrable function on M$M$. In the terminology of W.N. Everitt and M. Giertz, the differential expression L$L$ is said to be separated in L²(M)$L^2\left(M\right)$ if for all u∈L²(M)$u\in L^2\left(M\right)$ such that Lu∈L²(M)$Lu\in L^2\left(M\right)$, we have qu∈L²(M)$qu\in L^2\left(M\right)$. We give sufficient conditions for L$L$ to be separated in L²(M)$L^2\left(M\right)$.     

Twistor forms on Riemannian products

We study twistor forms on products of compact Riemannian manifolds and show that they are defined by Killing forms on the factors. The main result of this note is a necessary step in the classification of compact Riemannian manifolds with non-generic holonomy carrying twistor forms.     

Laplacian eigenvalue functionals and metric deformations on compact manifolds

In this paper, we investigate critical points of the eigenvalues of the Laplace operator considered as functionals on the space of Riemannian metrics or a conformal class of metrics on a compact manifold. We introduce a natural notion of the critical metric of such a functional and obtain necessary and sufficient conditions for a metric to be critical. We derive specific consequences concerning possible locally maximizing metrics. We also characterize critical metrics of the ratio of two consecutive eigenvalues.     

Renormalised Chern–Weil forms associated with families of Dirac operators

We provide local expressions for Chern–Weil type forms built from superconnections associated with families of Dirac operators previously investigated in [S. Scott, Zeta–Chern forms and the local family index theorem, Trans. Amer. Math. Soc. (in press). arXiv: math.DG/0406294] and later in [S. Paycha, S. Scott, Chern–Weil forms associated with superconnections, in: B. Booss-Bavnbeck, S. Klimek, M. Lesch, W. Zhang (Eds.), Analysis, Geometry and Topology of Elliptic Operators, World Scientific, 2006].     

Elliptic Equation with Singular Terms on Riemannian Manifold

In this paper we deal with the following equation: on a three-dimensional Riemannian manifold . We assume that the volume of Σ, the norm , and are small enough. Using a rather simple argument we show the existence of solution to the problem. Dedicated to Gosia and Basia.     

Brownian motion on manifolds with time-dependent metrics and stochastic completeness

The Brownian motion on a Riemannian manifold is a stochastic process such that the heat kernel is the density of the transition probability. If the total probability of the particle being found in the state space is constantly 1, then the Brownian motion is called stochastically complete. For manifolds with time-dependent metrics, the heat equation should be modified. With the modified heat equation, we study the Brownian motion on manifolds with time-dependent metrics and find conditions on metrics and the volume growth for stochastic completeness.     

On singular Lagrangian underlying the Schrödinger equation

We analyze the properties that manifest Hamiltonian nature of the Schrödinger equation and show that it can be considered as originating from singular Lagrangian action (with two second class constraints presented in the Hamiltonian formulation). It is used to show that any solution of the Schrödinger equation with time independent potential can be presented in the form , where the real field ?(t,xi) is some solution of nonsingular Lagrangian theory being specified below. Preservation of probability turns out to be the energy conservation law for the field ?. After introduction the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics.     

Wave operators for the Schrödinger equation with strongly singular short-range potentials

We use a semigroup positivity preserving to prove asymptotic completeness of the wave operators in many cases when they exist.     

On the Origin of the BV Operator on Odd Symplectic Supermanifolds

Differential forms on an odd symplectic manifold form a bicomplex: one differential is the wedge product with the symplectic form and the other is de Rham differential. In the corresponding spectral sequence the next differential turns out to be the Batalin–Vilkoviski operator.     

Superintegrable potentials on 3D Riemannian and Lorentzian spaces with nonconstant curvature

A quantum sl(2,ℝ) coalgebra (with deformation parameter z) is shown to underly the construction of a large class of superintegrable potentials on 3D curved spaces, that include the nonconstant curvature analog of the spherical, hyperbolic, and (anti-)de Sitter spaces. The connection and curvature tensors for these “deformed“ spaces are fully studied by working on two different phase spaces. The former directly comes from a 3D symplectic realization of the deformed coalgebra, while the latter is obtained through a map leading to a spherical-type phase space. In this framework, the nondeformed limit z → 0 is identified with the flat contraction leading to the Euclidean and Minkowskian spaces/potentials. The resulting Hamiltonians always admit, at least, three functionally independent constants of motion coming from the coalgebra structure. Furthermore, the intrinsic oscillator and Kepler potentials on such Riemannian and Lorentzian spaces of nonconstant curvature are identified, and several examples of them are explicitly presented.     

On Nth-order rogue wave solution to the generalized nonlinear Schrödinger equation

In this Letter, the generalized nonlinear Schrödinger (GNLS) equation is investigated by Darboux matrix method. A generalized Darboux transformation (DT) of the GNLS equation is constructed with the help of the gauge transformation for an Ablowitz–Kaup–Newell–Segur (AKNS) type GNLS spectral problem, from which a unified formula of Nth-order rogue wave solution to the GNLS equation is given. In particular, the first and second-order rogue wave solutions to the GNLS equation are explicitly illustrated through some figures.     

The influence of inelastic πω channel on pion form factor at s<(mω + mπ)2

It is shown that F_π(s) can be calculated in a model independent way if one knows the phase δ₁ and the inelasticity η of the p-wave ππ scattering and also F_π and the form factor of the γ¹ → π°ω transition for s> (m_ω + m_π)². The correction on F_π(s) for s< (m_ω + m_π)² due to the πω state with a strong ?′(1250) allows to explain the discrepancy between ?-dominance predictions and the experimental data.     

Regularity of the density of states in the Anderson model on a strip for potentials with singular continuous distributions

We derive regularity properties for the density of states in the Anderson model on a one-dimensional strip for potentials with singular continuous distributions. For example, if the characteristic function is infinitely differentiable with bounded derivatives and together with all its derivatives goes to zero at infinity, we show that the density of states is infinitely differentiable.