We examine the Bloch-Peierls-Berry dynamics under a classical nonequilibrium dynamical formulation. In this formulation all coordinates in phase space formed by the position and crystal momentum space are treated on equal footing. Explicitly demonstrations of the no (naive) Liouville theorem and of the validity of Darboux theorem are given. The explicit equilibrium distribution function is obta...

We show that Bloch oscillations are possible in dissipative optical waveguide lattices with a linearly varying propagation constant. These oscillations occur in spite of the fact that the Bloch wave packet experiences coupling gain and (or) loss. Experimentally, this process can be observed in different settings, such as in laser arrays and lattices of semiconductor optical amplifiers. In addit...

In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identi...

Let ϕ(z) = (ϕ 1 (z),...,ϕ n (z)) be a holomorphic self-map of D n and ψ(z) a holomorphic function on D n , where D n is the unit polydiscs of C n. Let 0 < α, β < 1, we compute the essential norm of a weighted composition operator ψC ϕ between α-Bloch space Ꮾ α (D n) and β-Bloch space Ꮾ β (D n).

Let X be a smooth, projective algebraic variety over an algebraically closed field k, and let H(X) denote a Weil cohomology theory. The existence of algebraic cycles on X×X inducing as correspondences the various Künneth projectors π : H(X) → H (X) is one of the standard conjectures of Grothendieck, [8], [9]. It is known in general only for the cases i = 0, 1, 2d − 1, 2d where d = dimX. The pur...

We give two positive results on the inverse spectral problem for simply connected analytic domains with discrete symmetries. First, we consider domains with one mirror isometry and with an invariant bouncing ball orbit whose orientation is reversed by the symmetry. We prove that such domains are determined by their Dirichlet spectra among other domains in that class. Second, we consider domains...

1 Raison d’être 1 1.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Schrödinger Operator with Periodic Potential . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Direct Integral Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Bloch Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

Bloch oscillation in electrically biased semiconductor superlattices offers broadband terahertz gain from DC up to the Bloch frequency and can provide a basis for solid-state electronic oscillators operating at 10 times the frequency of existing devices. To circumvent the inherent instability of the electrically biased doped superlattices to the formation of static or dynamic electric field dom...

Bloch oscillations of cold atoms in two-dimensional optical lattices are studied. The cases of separable and nonseparable potentials are compared by simulating the wave-packet dynamics. For these two classes of optical potential, the Bloch oscillations were found to be qualitatively the same in the case of a weak static field but fundamentally different in the case of a strong field. In additio...