198,026 research outputs found
The PT-symmetric brachistochrone problem, Lorentz boosts and non-unitary operator equivalence classes
The PT-symmetric (PTS) quantum brachistochrone problem is reanalyzed as
quantum system consisting of a non-Hermitian PTS component and a purely
Hermitian component simultaneously. Interpreting this specific setup as
subsystem of a larger Hermitian system, we find non-unitary operator
equivalence classes (conjugacy classes) as natural ingredient which contain at
least one Dirac-Hermitian representative. With the help of a geometric analysis
the compatibility of the vanishing passage time solution of a PTS
brachistochrone with the Anandan-Aharonov lower bound for passage times of
Hermitian brachistochrones is demonstrated.Comment: 12 pages, 2 figures, strongly extended versio
Exterior domain problems and decomposition of tensor fields in weighted Sobolev spaces
The Hodge decompOsition is a useful tool for tensor analysis on compact manifolds with boundary. This paper aims at generalising the decomposition to exterior domains G ⊂ IR n. Let L 2a Ω k(G) be the space weighted square integrable differential forms with weight function (1 + |χ|²)a, let d a be the weighted perturbation of the exterior derivative and δ a its adjoint. Then L 2a Ω k(G) splits into the orthogonal sum of the subspaces of the d a-exact forms with vanishing tangential component on the boundary, of δ a-coexact forms with vanishing normal component, and harmonic forms, in the sense of d a λ
The general relativistic thin disc evolution equation
In the classical theory of thin disc accretion discs, the constraints of mass
and angular momentum conservation lead to a diffusion-like equation for the
turbulent evolution of the surface density. Here, we revisit this problem,
extending the Newtonian analysis to the regime of Kerr geometry relevant to
black holes. A diffusion-like equation once again emerges, but now with a
singularity at the radius at which the effective angular momentum gradient
passes through zero. The equation may be analysed using a combination of WKB,
local techniques, and matched asymptotic expansions. It is shown that imposing
the boundary condition of a vanishing stress tensor (more precisely the
radial-azimuthal component thereof) allows smooth stable modes to exist
external to the angular momentum singularity, the innermost stable circular
orbit, while smoothly vanishing inside this location. The extension of the disc
diffusion equation to the domain of general relativity introduces a new tool
for numerical and phenomenolgical studies of accretion discs, and may prove to
be a useful technique for understanding black hole X-ray transients.Comment: 7 Pages, 1 figure. Accepted for publication in MNRAS. Revised version
corrects minor typos in equations (64) and (66) of original, otherwise
unaltere
Enhancement of field renormalization in scalar theories via functional renormalization group
The flow equations of the Functional Renormalization Group are applied to the
O(N)-symmetric scalar theory, for N=1 and N=4, in four Euclidean dimensions,
d=4, to determine the effective potential and the renormalization function of
the field in the broken phase. In our numerical analysis, the infrared limit,
corresponding to the vanishing of the running momentum scale in the equations,
is approached to obtain the physical values of the parameters by extrapolation.
In the N=4 theory a non-perturbatively large value of the physical
renormalization of the longitudinal component of the field is observed. The
dependence of the field renormalization on the UV cut-off and on the bare
coupling is also investigated.Comment: 20 pages, 7 figures. To appear in Physical Review
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