A family of Poisson non-compact symmetric spaces

We study Poisson symmetric spaces of group type with Cartan subalgebra “adapted” to the Lie cobracket.     

Higher local conservation laws for nonlinear sigma models on symmetric spaces

We show that for the classical two-dimensional nonlinear -model on a Riemannian symmetric space of dimensionm and rankp, there existp independent series of higher local conservation laws, and we reduce the field equations of the model to a system of nonlinear partial differential equations possessing an associated Lax pair and involvingm+p independent variables.Dedicated to the Memory of our Colleague and Friend Jorge André SwiecaWork partially done under DFG contract Schr 4/5     

On the spectrum of Kaluza-Klein theories with non-compact internal spaces

We investigate the conditions under which Kaluza-Klein theories with non-compact internal spaces admit a discrete spectrum.     

The phase of the Riemann zeta function

We, offer an alternative interpretation of the Riemann zeta functionζ(s) as a scattering amplitude and its nontrivial zeros as the resonances in the scattering amplitude. We also look at several different facets of the phase of theζ function. For example, we show that the smooth part of theζ function along the line of the zeros is related to the quantum density of states of an inverted oscillator. On the other hand, for ℜs>1/2, we show that the memory of the zeros fades only gradually through a Lorentzian smoothing of the delta functions. The corresponding trace formula for ℜs≫1 is shown to be of the same form as generated by a one-dimensional harmonic oscillator in one direction along with an inverted oscillator in the transverse direction. Quite remarkably for this simple model, the Gutzwiller trace formula can be obtained analytically and is found to agree with the quantum result.     

The zeta function of the quartic oscillator

We investigate the levels of the quartic oscillator ($\text{?}\text{d}{}^2\text{d}\text{q}{}^2\text{+ q}{}^4$) by means of the associated zeta function. By using several versions of the semiclassical approach, we prove exact results concerning the locations of the real zeros and of the poles of that function, the residues of which are essentially the coefficients of the semiclassical expansion of the Bohr-Sommerfeld quantization rule. We also compute the derivative of the zeta function at the origin, by relating it to the Fredholm determinant of the operator. These results translate as sum rules for the eigenvalues and thus are helpful for their precise computation, and a further analysis allows one to split them into separate rules for even and odd levels; the extension to higher anharmonic oscillators ($\text{?}\text{d}{}^2\text{d}\text{q}{}^2\text{+ q}{}^{\mathrm{2M}}$) is immediate. Finally we propose an asymptotic formula for large negative arguments of the zeta function, based on the nature and location of the complex singularities of the partition function of the operator $\text{(}\text{?}\text{d}{}^2\text{d}\text{q}{}^2\text{+ q}{}^4\text{)}{}^{\mathrm{<mtext>3</mtext><mtext>4</mtext>}}$.     

The spinor heat kernel in maximally symmetric spaces

The heat kernelK(x, x, t) of the iterated Dirac operator on anN-dimensional simply connected maximally symmetric Riemannian manifold is calculated. On the odd-dimesional hyperbolic spacesK is a Minakshisundaram-DeWitt expansion which terminates to the coefficienta _N–1)/2 and is exact. On the odd spheres the heat kernel may be written as an image sum of WKB kernels, each term corresponding to a classical path (geodesic). In the even dimensional case the WKB approximation is not exact, but a closed form ofK is derived both in terms of (spherical) eigenfunctions and of a sum over classical paths. The spinor Plancherel measure () and function in the hyperbolic case are also calculated. A simple relation between the analytic structure of onH ^N and the degeneracies of the Dirac operator onS ^N is found.     

Quantum zeta function for perturbed cat maps

The behavior of semiclassical approximations to the spectra of perturbed quantum cat maps is examined as the perturbation parameter brings the corresponding classical system into the nonhyperbolic regime. The approximations are initially accurate but large errors are found to appear in the traces and in the coefficients of the characteristic polynomial after nonhyperbolic structures appear. Nevertheless, the eigenvalues obtained from them remain accurate up to large perturbations. (c) 1995 American Institute of Physics.     

Conformational completion of Lorentz symmetric spaces

Every simply connected Lorentz symmetric space of dimensionn3 has a symmetric quotient which can be conformally imbedded in a quadric of the projective space ⁿ⁺¹ (R).     

A variational principle for actions on symmetric symplectic spaces

We present a definition of generating functions of canonical relations, which are real functions on symmetric symplectic spaces, discussing some conditions for the presence of caustics. We show how the actions compose by a neat geometrical formula and are connected to the hamiltonians via a geometrically simple variational principle which determines the classical trajectories, discussing the temporal evolution of such “extended hamiltonians” in terms of Hamilton–Jacobi-type equations. Simplest spaces are treated explicitly.     

Functional equation for dynamical zeta functions of Milnor-Thurston type

A Milnor-Thurston type dynamical zeta function _L (Z) is associated with a family of maps of the interval (–1, 1). Changing the direction of time produces a new zeta function _L (Z). These zeta functions satisfy a functional equation _L (Z) _L (Z)= ₀(Z) (where amounts to sign changes and, generically,₀1). The functional equation has non-trivial implications for the analytic properties of _L (Z).     

The local geometry of compact homogeneous Lorentz spaces

In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.     

Local conserved currents for nonlinear σ-models on symmetric spaces

A general method is presented to construct an infinite series of conserved local charges for a large class of two-dimensional nonlinear σ-models on symmetric spaces. The conservation laws are derived from a couple of first order Ricatti differential equations using the dual symmetry of σ-models on symmetric spaces. The method is exemplified for the case of σ-models on Grassmann manifolds.     

Estimates and extremals for zeta function determinants on four-manifolds

LetA be a positive integral power of a natural, conformally covariant differential operator on tensor-spinors in a Riemannian manifold. Suppose thatA is formally self-adjoint and has positive definite leading symbol. For example,A could be the conformal Laplacian (Yamabe operator)L, or the square of the Dirac operator     

Stability of higher-dimensional Einstein-Yang-Mills theories on symmetric spaces

We study (4 + d)-dimensional Einstein-Yang-Mills theories with arbitrary gauge groups, G_YM. The theory is compactified on a d-dimensional symmetric coset space $\text{G}\text{H}$ with a symmetric, topologically non-trivial classical gauge field, embedded in an H-subgroup of the Yang-Mills group. These theories are known to be classically stabilized by gravity if G_YM = H, $\text{G}\text{H}$ is a sphere and d ≠ 3. We study classical instabilities caused by embedding H in a larger gauge group. The small fluctuation spectrum is completely calculable, and leads to a stability condition. For two-dimensional spheres this condition is precisely the Brandt-Neri stability condition for non-abelian monopole fields. For four-spheres we find stability for SU(2) instantons embedded in arbitrary gauge groups and we reproduce the fluctuation spectrum around instantons. For higher-dimensional spheres the stable solutions of this type are completely classified, and occur only for d = 5, 6, 8, 9, 10, 12 and 16. The results show a remarkable agreement with expected topological stability. We also give a few examples with other symmetric spaces, such as CP_n, where the stability criterion appears less restrictive.     

Formal iteration scheme for scalar fields in some spherically symmetric spaces

The characteristic initial-value problem for the scalar wave equation in some spherically symmetrical geometries, including Reissner-Nordström exterior geometry, is considered. A modified Picard iteration making explicit use of the (known) static solutions yields a formula for the fields. The formulas extend in to the outer horizon, and from them are obtained useful expressions for the reflection and transmission amplitudes in terms of the static solutions.     

Locally symmetric riemannian spaces and solutions of Yang-Mills equations

We give an explicit construction for all classical solutions of Euclidean SU(2) Yang-Mills equations which can be obtained by reduction of conformally flat locally symmetric Riemannian spaces.Address after 1 April 1987: Institut für Theoretische Physik, Universität Karlsruhe, D-7500 Karlsruhe 1, West Germany.     

局域单PT对称势中多级孤子的传输特性研究

具有Parity-Time(PT)对称性的光学系统是近年提出的一种新型光学结构,在光开关、光子信息处理器件等方面具有潜在的应用。研究了局域单PT对称光学系统中双级孤子和三级孤子的存在范围与稳定特性。研究结果表明:多级孤子存在一临界传播常数,传播常数对应了线性模下的本征值。双级孤子和三级孤子仅能在较深的PT对称势中才能稳定地传输。多级孤子的稳定性的范围随着PT对称势的深度增加而扩大;孤子的能量随着传播常数的增大而增加,但随着PT对称势的调制深度的增加而减少。     

A family of poisson structures on hermitian symmetric spaces

We investigate the compatibility of symplectic Kirillov-Kostant-Souriau structure and Poisson-Lie structure on coadjoint orbits of semisimple Lie group. We prove that they are compatible for an orbit compact Lie group iff the orbit is hermitian symmetric space. We prove also the compatibility statement for non-compact hermitian symmetric space. As an example we describe a structure of symplectic leaves onCP ⁿ for this family. These leaves may be considered as a perturbation of Schubert cells. Possible applications to infinite-dimensional examples are discussed.     

New examples of indefinite hyper-Kähler symmetric spaces

Following the approach to pseudo-Riemannian symmetric spaces developed in [I. Kath, M. Olbrich, On the structure of pseudo-Riemannian symmetric spaces, arXiv:math.DG/0408249, 2004] we exhibit examples of indefinite hyper-Kähler symmetric spaces with non-abelian holonomy. Moreover, we classify indecomposable hyper-Kähler symmetric spaces whose metric has signature (4,4n)$(4,4n)$. Such spaces exist if and only if n∈{0,1,3}$n\in \{0,1,3\}$.