High order WKB prediction of the energy splitting in the symmetric double well potential

The accuracy of the WKB approximation when predicting the energy splitting of bound states in a double well potential is the main subject of this paper. The splitting of almost degenerate energy levels below the top of the barrier results from the tunneling and is thus supposed to be exponentially small. By using the standard WKB quantization we deduce an analytical formula for the energy splitting, which is the usual Landau formula with additional quantum corrections. We also examine the accuracy of our and Landau formula numerically for the case of the symmetric double well quartic potential.Comment: 8 pages, 2 figures, 1 table, PTP LaTeX style, to be published in the proceedings of the conference/summer school 'Let's Face Chaos through Nonlinear Dynamics', Maribor, Slovenia, June/July 1999, eds. M. Robnik et al., Prog. Theor. Phys. Suppl. (Kyoto) 139 (2000

Spin diffusion from an inhomogeneous quench in an integrable system

Generalised hydrodynamics predicts universal ballistic transport in integrable lattice systems when prepared in generic inhomogeneous initial states. However, the ballistic contribution to transport can vanish in systems with additional discrete symmetries. Here we perform large scale numerical simulations of spin dynamics in the anisotropic Heisenberg $XXZ$ spin $1/2$ chain starting from an inhomogeneous mixed initial state which is symmetric with respect to a combination of spin-reversal and spatial reflection. In the isotropic and easy-axis regimes we find non-ballistic spin transport which we analyse in detail in terms of scaling exponents of the transported magnetisation and scaling profiles of the spin density. While in the easy-axis regime we find accurate evidence of normal diffusion, the spin transport in the isotropic case is clearly super-diffusive, with the scaling exponent very close to $2/3$, but with universal scaling dynamics which obeys the diffusion equation in nonlinearly scaled time.Comment: 8 pages, 7 figures, version as accepted by Nature Communication

Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm

Let \F_q ($q=p^r$) be a finite field. In this paper the number of irreducible polynomials of degree $m$ in \F_q[x] with prescribed trace and norm coefficients is calculated in certain special cases and a general bound for that number is obtained improving the bound by Wan if $m$ is small compared to $q$. As a corollary, sharp bounds are obtained for the number of elements in \F_{q^3} with prescribed trace and norm over \F_q improving the estimates by Katz in this special case. Moreover, a characterization of Kloosterman sums over \F_{2^r} divisible by three is given generalizing the earlier result by Charpin, Helleseth, and Zinoviev obtained only in the case $r$ odd. Finally, a new simple proof for the value distribution of a Kloosterman sum over the field \F_{3^r}, first proved by Katz and Livne, is given.Comment: 21 pages; revised version with somewhat more clearer proofs; to appear in Acta Arithmetic