Heisenberg Group

We consider the biharmonicity condition for maps between Riemannian manifolds (see [BK]), and study the non-geodesic biharmonic curves in the Heisenberg group H_3. First we prove that all of them are helices, and then we obtain explicitly their parametric equations.

Quantized contact transformations are Toeplitz operators over a contact manifold $(X,\alpha)$ of the form $U_{\chi} = \Pi A \chi \Pi$, where $\Pi : H^2(X) \to L^2(X)$ is a Szego projector, where $\chi$ is a contact transformation and where $A$ is a pseudodifferential operator over $X$. They provide a flexible alternative to the Kahler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine $ind(U_{\chi})$ when the principal symbol is unitary, or equivalently to determine whether $A$ can be chosen so that $U_{\chi}$ is unitary. We show that the answer is yes in the case of quantized symplectic torus automorphisms $g$---by showing that $U_g$ duplicates the classical transformation laws on theta functions. Using the Cauchy-Szego kernel on the Heisenberg group, we calculate the traces on theta functions of each degree N. We also study the quantum dynamics generated by a general q.c.t. $U_{\chi}$, i.e. the quasi-classical asymptotics of the eigenvalues and eigenfunctions under various ergodicity and mixing hypotheses on $\chi.$ Our principal results are proofs of equidistribution of eigenfunctions $\phi_{Nj}$ and weak mixing properties of matrix elements $(B\phi_{Ni}, \phi_{Nj})$ for quantizations of mixing symplectic maps.

Math. Proc. Camb. Phil. Soc. (2001), 131, 487 Printed in the United Kingdom c© 2001 Cambridge Philosophical Society ... Hardy's Theorem for simply connected nilpotent Lie groups ... By EBERHARD KANIUTH Fachbereich Mathematik/Informatik, Universität Paderborn, D-33095 ...

In this manuscript, we develope the theory of harmonic analysis on the Heisenberg group G of high dimension. We investigate the theta functions and the Weil representation related to this Heisenberg group and describe the connection among them.

The Schr\"odinger-Robertson inequality for relativistic position and momentum operators X^\mu, P_\nu, \mu, \nu = 0,1,2,3, is interpreted in terms of Born reciprocity and `non-commutative' relativistic phase space geometry. For states which saturate the Schr\"odinger-Robertson inequality, a typology of semiclassical limits is pointed out, characterised by the orbit structure within its unitary irreducible representations, of the full invariance group of Born reciprocity, the so-called `quaplectic' group U(3,1)xH(3,1) (the semi-direct product of the unitary relativistic dyamical symmetry U(3,1) with the Weyl-Heisenberg group H(3,1)). The example of the `scalar' case, namely the relativistic oscillator, and associated multimode squeezed states, is treated in detail. In this case,it is suggested that the semiclassical limit corresponds to the separate emergence of space-time and matter, in the form of the stress-energy tensor, and the quadrupole tensor, which are in general reciprocally equivalent.

We study coercive inequalities on finite dimensional metric spaces with probability measures which do not have volume doubling property. This class of inequalities includes Poincar\'e and Log-Sobolev inequality. Our main result is proof of Log-Sobolev inequality on Heisenberg group equipped with either heat kernel measure or "gaussian" density build from optimal control distance. As intermediate results we prove so called U-bounds.

We establish several versions of Hardy's theorem for the Fourier transform on the Heisenberg group. Let \(\hat f (\lambda)\) be the Fourier transform of a function f on \(H^n\) and assume \(\hat f (\lambda)^\ast \hat f (\lambda) \leq c \hat p_{2b} (\lambda)\) where \(p_s\) is the heat kernel associated to the sublaplacian. We show that if \(|f(z,t)| \leq c p_a (z,t)\) then \(f=0\) whenever \(a < b\) . When \(a \geq b\) we replace the condition on f by \(|f^\lambda (z)| \leq c p_a^\lambda (z)\) where \(f^\lambda (z)\) is the Fourier transform of f in the t-variable. Under suitable assumptions on the ‘spherical harmonic coefficients’ of \(\hat f (\lambda)\) we prove: (i) \(f^\lambda (z)=c(\lambda) p^\lambda_a (z)\) when a=b; (ii) when a > b there are infinitely many linearly independent functions f satisfying both conditions on \(f^\lambda\) and \(\hat f (\lambda)\) .

We extend results of the Pompeiu problem on the Heisen- berg group Hn from spheres to complex ellipsoids. These results also tell us what happens for spheres and complex ellipsoids on the anisotropic Heisenberg group, Hna. The results for L 2 , Lp, and L1 have the same character as previous results for spheres on Hn. However, when moving to L1 and including rotations, we maintain the result from Euclidean space that only one complex ellipsoid is needed.

The Jacobi group is the semi-direct product of the symplectic group and the Heisenberg group. The Jacobi group is an important object in the frame- work of quantum mechanics, geometric quantization and optics. In this paper, we study the Weil representations of the Jacobi group and their properties. We also provide their applications to the theory of automorphic forms on the Jacobi group and representation theory of the Jacobi group.

For a class of discrete quasi-periodic Schrödinger operators defined by covariant representations of the rotation algebra, a lower bound on phase-averaged transport in terms of the multifractal dimensions of the density of states is proven. This result is established under a Diophantine condition on the incommensuration parameter. The relevant class of operators is distinguished by invariance with respect to symmetry automorphisms of the rotation algebra. It includes the critical Harper (almost-Mathieu) operator. As a by-product, a new solution of the frame problem associated with Weyl-Heisenberg-Gabor lattices of coherent states is given.

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold endowed with the Webster metric hence formulate a version of the CR Yamabe problem for CR manifolds-with-boundary. This is shown to be a nonlinear subelliptic problem of variational origin.