Eingenvalues of the Dirac operator on compact Kähler manifolds

Kählerian twistor operators are introduced to get lower bounds for the eigenvalues of the Dirac operator on compact spin Kähler manifolds. In odd complex dimensions, manifolds with the smallest eigenvalues are characterized by an over determined system of differential equations similar to the Riemannian case. In these dimensions, we show the existence of a unique natural Kählerian twistor operator. It is also proved that, on a Kähler manifold with nonzero scalar curvature, the space of Riemannian twistor-spinors is trivial.This work has been partially supported by the EEC programme GADGET Contract Nr. SC1-0105     

Geodesic symmetries and invariant star products on Kähler symmetric spaces

Starting from work by F. A. Berezin, an earlier paper by the author obtained an invariant star product on every nonexceptional symmetric Kähler space. This would be a generalization to those spaces of the star product on ²ⁿ corresponding to Wick quantization. In this Letter we consider, via geometric quantization, the unitary operators corresponding to geodesic symmetries, and we define a Weyl quantization (first defined by Berezin on rank 1 spaces) in a way similar to the way in which the Weyl quantization can be obtained from the Wick quantization on ²ⁿ . We then calculate every Hochschild 2-cochain of another invariant star product, equivalent to the Wick one, which would be a generalization to those spaces of the Moyal star product on ²ⁿ . M. Cahen and S. Gutt have already provided a theorem of existence and essential unicity of an invariant star product on every irreducible Kähler symmetric space.     

On the eigenvalue problem for the schrödinger operator with a confining potential

A new method for the computation of the eigenvalues of the Schrödinger operator proposed recently is applied here to the case of a confining potential. It is shown that one can draw conclusions similar to those in the case of short-range potentials discussed in previous work.Dedicated to Academician Václav Votruba on the occasion of his seventieth birthday.     

Spectral series of the Schrödinger operator with delta-potential on a three-dimensional spherically symmetric manifold

The spectral series of the Schrödinger operator with a delta-potential on a threedimensional compact spherically symmetric manifold in the semiclassical limit as h → 0 are described.     

The Dynamics of the Energy of a Kähler Class

The energy of a Kähler class, on a compact complex manifold (M,J) of Kähler type, is the infimum of the squared L²-norm of the scalar curvature over all Kähler metrics representing the class. We study general properties of this functional, and define its gradient flow over all Kähler classes represented by metrics of fixed volume. When besides the trivial holomorphic vector field of (M,J), all others have no zeroes, we extend it to a flow over all cohomology classes of fixed top cup product. We prove that the dynamical system in this space defined by the said flow does not have periodic orbits, that its only fixed points are critical classes of a suitably defined extension of the energy function, and that along solution curves in the Kähler cone the energy is a monotone function. If the Kähler cone is forward invariant under the flow, solutions to the flow equation converge to a critical point of the class energy function. We show that this is always the case when the manifold has a signed first Chern class. We characterize the forward stability of the Kähler cone in terms of the value of a suitable time dependent form over irreducible subvarieties of (M,J). We use this result to draw several geometric conclusions, including the determination of optimal dimension dependent bounds for the squared L²-norm of the scalar curvature functional.Acknowledgement We would like to thank Nicholas Buchdahl for helpful conversations leading us to several improvements of an earlier version of the article, including the correction of two improper assertions.     

Renormalization properties of Bosonic non-linear σ-models built on compact homogeneous Kähler manifolds

The non-linear bosonic σ-models built on compact Kähler homogeneous manifolds are parametrized in such a way that multiplicative renormalizability holds, to all orders of perturbation theory. Moreover, the fields are not renormalized. The essential ingredients of the proof are the homogeneity of the space and the existence of a charge Y that separates the fields in φ and φ̄.     

First Eigenvalue of the Dirac Operator for a Kähler Manifold of Even Complex Dimension: The Limiting Case

K. D. Kirchberg has given a minoration of the absolute value of the eigenvalues of the Dirac operator for a compact Kähler spin manifold (W,g) with positive scalar curvature and has introduced, in this context, the notion of Kähler twistor-spinor. We prove here that if dim_C W = p 4 is even, in the limiting case, (W, g) is the Kähler product of an odd-dimensional limiting case compact Kähler spin manifold of complex dimension (p-1), by a flat Kähler manifold which is a compact quotient of C.     

A local index theorem for families of\bar \partial -operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaceson punctured Riemann surfaces and a new Kähler metric on their moduli spaces

We prove a local index theorem for families of \(\bar \partial \) -operators on Riemann surfaces of type (g, n), i.e. of genusg withn>0 punctures. We calculate the first Chern form of the determinant line bundle on the Teichmüller spaceT _g,n endowed with Quillen's metric (where the role of the determinant of the Laplace operators is played by the values of the Selberg zeta function at integer points). The result differs from the case of compact Riemann surfaces by an additional term, which turns out to be the Kähler form of a new Kähler metric on the moduli space of punctured Riemann surfaces. As a corollary of this result we derive, for instance, an analog of Mumford's isomorphism in the case of the universal curve.     

On the number of negative eigenvalues for a Schrödinger operator with magnetic field

We consider the Schrödinger operator with magnetic field $$H = \sum\limits_{j = 1}^n {\left( {\frac{1}{i}\frac{\partial }{{\partial x_j }} - a_j } \right)^2 + Vin\mathbb{R}^n .} $$ Under certain conditions on the magnetic fieldB=curla, we generalize the Fefferman—Phong estimates (Bull. A. M. S.9, 129–206 (1983)) on the number of negative eigenvalues for ?Δ+V to the operatorH. Upper and lower bounds are established. Our estimates incorporate the contribution from the magnetic field. The conditions onB in particular are satisfied if the magnetic potentialsa _j (x) are polynomials.     

Effect of impurity on the absorption of a parabolic quantum dot with including Rashba spin–orbit interaction

In this paper, the influence of impurity parameters on the electron energy spectrum and absorption coefficients in a parabolic quantum dot and in the presence of Rashba spin–orbit interaction subjected to a perpendicular magnetic field is studied. The impurity potential is approximated by a Gaussian form. We have shown that in the both cases of a repulsive and attractive Gaussian impurity, the absorption coefficients are strongly affected by the decay length. These coefficients show blue (red) shift as the decay length of repulsive (attractive) impurity is increased. The dependence of the absorption coefficients on the impurity position is also examined for different polarizations. Our results show that the absorption coefficient has local maximum (minimum) for a given value of impurity position for Y-polarized (X-polarized) light.     

Spectral series of the Schrödinger operator in a thin waveguide with a periodic structure. 2. Closed three-dimensional waveguide in a magnetic field

In the paper, which is the second part of the paper by J. Brüning, S. Dobrokhotov, S. Sekerzh-Zenkovich, T. Tudorovskiy, “Spectral series of the Schrödinger operator in thin waveguides with a periodic structure. 1,” Russ. J. Math. Phys. 13 (4), 401–420 (2006), using the adiabatic approximation, diverse quantum states of the stationary Schrödinger equation for a particle in a thin waveguide in a magnetic field are constructed. The problems of “destruction” of the adiabatic approximation as the value of energy increases and of replacing this approximation by the approximation of V. P. Maslov’s theory of complex germ (the complex WKB method) are studied.     

On coding of a quantum source of states with a finite frequency band: Quantum analogue of the Kotel’nikov theorem on sampling

An example of coding a source of quantum states with a finite frequency band W and finite exit power not exceeding ～(?W)W is given. The number of classical information bits that can be coded in the quantum states generated by such a source per unit time is C=W. Such a source is minimal in the sense that the filling factor for each of the orthogonal single-particle modes constituting N=WT-photon vector in time window 2T is equal to 1. This result can be treated as a quantum analogue of the Kotel’nikov theorem on sampling for classical signals     

Curing One Discontinuity With Another: Comment on S. K. Ghosal Et Al., “On the Anisotropy of the Speed of Light on a Rotating Platform”

No Heading A recent paper by Ghosal, Raychaudhuri, Chowdhury, and Sarker (GRCS in the following) discussed a seven-years old argument in relativity concerning the speed of light as measured by an observer on board a rotating disk. The argument, put forward by the present author, contains the theoretical proof of an anisotropy in the speed of light in a reference frame comoving with the edge of a rotating disk even in the limit of zero acceleration. This conclusion challenges the internal consistency of the relativistic theories and undermines the basic tenet of the conventionality thesis of relativistic simultaneity. GRCS believe to be able to resolve the issue by recasting the original argument in the Galilean world and thereby exposing the weak points of the reasonings leading to the fallacy. We will defend the original argument and show that the treatment proposed by GRCS is as bad as the illness it asserts to cure.     

Different effect of NiMnCo or FeNiCo on the growth of type-Ⅱa large diamonds with Ti/Cu as nitrogen getter

In order to synthesize high-quality type-Ⅱa large diamond, the selection of catalyst is very important, in addition to the nitrogen getter. In this paper, type-IIa large diamonds are grown under high pressure and high temperature(HPHT) by using the temperature gradient method(TGM), with adopting Ti/Cu as the nitrogen getter in Ni_(70)Mn_(25)Co_5(abbreviated as NiMnCo) or Fe_(55)Ni_(29)Co_(16)(abbreviated FeNiCo) catalyst. The values of nitrogen concentration(N_c) in both synthesized high-quality diamonds are less than 1 ppm, when Ti/Cu(1.6 wt%) is added in the FeNiCo or Ti/Cu(1.8 wt%) is added in the NiMnCo. The difference in solubility of nitrogen between both catalysts at HPHT is the basic reason for the different effect of Ti/Cu on eliminating nitrogen. The nitrogen-removal efficiency of Ti/Cu in the NiMnCo catalyst is less than in the FeNiCo catalyst. Additionally, a high-quality type-Ⅱa large diamond size of 5.0 mm is obtained by reducing the growth rate and keeping the nitrogen concentration of the diamond to be less than 1 ppm, when Ti/Cu(1.6 wt%) is added in the FeNiCo catalyst.     

Erratum to “Lower bounds for the eigenvalues of the basic Dirac operator on a Riemannian foliation” [J. Geom. Phys. 51 (2004) 166–182]

We correct some statements in the paper mentioned in the title.     

Dynamic magnetic behavior of the mixed spin （2, 5/2） Ising system with antiferromagnetic/antiferromagnetic interactions on a bilayer square lattice

Using the mean-field theory and Glauber-type stochastic dynamics, we study the dynamic magnetic properties of the mixed spin （2, 5/2） Ising system for the antiferromagnetic/antiferromagnetic （AFM/AFM） interactions on the bilayer square lattice under a time varying （sinusoidal） magnetic field. The time dependence of average magnetizations and the thermal variation of the dynamic magnetizations are examined to calculate the dynamic phase diagrams. The dynamic phase diagrams are presented in the reduced temperature and magnetic field amplitude plane and the effects of interlayer coupling interaction on the critical behavior of the system are investigated. We also investigate the influence of the frequency and find that the system displays richer dynamic critical behavior for higher values of frequency than that of the lower values of it. We perform a comparison with the ferromagnetic/ferromagnetic （FM/FM） and AFM/FM interactions in order to see the effects of AFM/AFM interaction and observe that the system displays richer and more interesting dynamic critical behaviors for the AFM/AFM interaction than those for the FM/FM and AFM/FM interactions.