57,690 research outputs found
The Riemann hypothesis for certain integrals of Eisenstein series
This paper studies the non-holomorphic Eisenstein series E(z,s) for the
modular surface, and shows that integration with respect to certain
non-negative measures gives meromorphic functions of s that have all their
zeros on the critical line Re(s) = 1/2. For the constant term of the Eisenstein
series it shows that all zeros are on the critical line for fixed y= Im(z) \ge
1, except possibly for two real zeros, which are present if and only if y > 4
\pi e^{-\gamma} = 7.0555+. It shows the Riemann hypothesis holds for all
truncation integrals with truncation parameter T \ge 1. For T=1 this proves the
Riemann hypothesis for a zeta function recently introduced by Lin Weng,
attached to rank 2 semistable lattices over the rationals.Comment: 23 pages; corrected residues of functions in theorem 1 and 2, added
reference; v3 small corrections, removed uncited references; v4 more small
correction
Slow quench dynamics of the Kitaev model: anisotropic critical point and effect of disorder
We study the non-equilibrium slow dynamics for the Kitaev model both in the
presence and the absence of disorder. For the case without disorder, we
demonstrate, via an exact solution, that the model provides an example of a
system with an anisotropic critical point and exhibits unusual scaling of
defect density and residual energy for a slow linear quench. We provide
a general expression for the scaling of () generated during a slow
power-law dynamics, characterized by a rate and exponent ,
from a gapped phase to an anisotropic quantum critical point in dimensions,
for which the energy gap for momentum
components () and for the rest components
() with : ().
These general expressions reproduce both the corresponding results for the
Kitaev model as a special case for and and the well-known
scaling laws of and for isotropic critical points for . We also
present an exact computation of all non-zero, independent, multispin
correlation functions of the Kitaev model for such a quench and discuss their
spatial dependence. For the disordered Kitaev model, where the disorder is
introduced via random choice of the link variables in the model's
Fermionic representation, we find that and () for a slow linear quench ending in the gapless
(gapped) phase. We provide a qualitative explanation of such scaling.Comment: 10 pages, 11 Figs. v
Neutrino-12C scattering in the ab initio shell model with a realistic three-body interaction
We investigate cross sections for neutrino-12C exclusive scattering and for
muon capture on 12C using wave functions obtained in the ab initio no-core
shell model. In our parameter-free calculations with basis spaces up to the 6
hbarOmega we show that realistic nucleon-nucleon interactions, like e.g. the
CD-Bonn, under predict the experimental cross sections by more than a factor of
two. By including a realistic three-body interaction, Tucson-Melbourne TM'(99),
the cross sections are enhanced significantly and a much better agreement with
experiment is achieved. At the same time,the TM'(99) interaction improves the
calculated level ordering in 12C. The comparison between the CD-Bonn and the
three-body calculations provides strong confirmation for the need to include a
realistic three-body interaction to account for the spin-orbit strength in
p-shell nuclei.Comment: 6 pages, 2 figure
Large basis ab initio shell model investigation of 9-Be and 11-Be
We are presenting the first ab initio structure investigation of the loosely
bound 11-Be nucleus, together with a study of the lighter isotope 9-Be. The
nuclear structure of these isotopes is particularly interesting due to the
appearance of a parity-inverted ground state in 11-Be. Our study is performed
in the framework of the ab initio no-core shell model. Results obtained using
four different, high-precision two-nucleon interactions, in model spaces up to
9\hbar\Omega, are shown. For both nuclei, and all potentials, we reach
convergence in the level ordering of positive- and negative-parity spectra
separately. Concerning their relative position, the positive-parity states are
always too high in excitation energy, but a fast drop with respect to the
negative-parity spectrum is observed when the model space is increased. This
behavior is most dramatic for 11-Be. In the largest model space we were able to
reach, the 1/2+ level has dropped down to become either the first or the second
excited state, depending on which interaction we use. We also observe a
contrasting behavior in the convergence patterns for different two-nucleon
potentials, and argue that a three-nucleon interaction is needed to explain the
parity inversion. Furthermore, large-basis calculations of 13-C and 11-B are
performed. This allows us to study the systematics of the position of the first
unnatural-parity state in the N=7 isotone and the A=11 isobar. The 11-B run in
the 9\hbar\Omega model space involves a matrix with dimension exceeding 1.1 x
10^9, and is our largest calculation so far. We present results on binding
energies, excitation spectra, level configurations, radii, electromagnetic
observables, and 10-Be+n overlap functions.Comment: 17 pages, 12 figures To be published in Phys. Rev. C Resubmitted
version. Minor change
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