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Register a new userOn the spectrum of the magnetohydrodynamic mean-field alpha^2-dynamo operator
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The existence of magnetohydrodynamic mean-field alpha^2-dynamos with spherically symmetric, isotropic helical turbulence function alpha is related to a non-self-adjoint spectral problem for a coupled system of two singular second order ordinary differential equations. We establish global estimates for the eigenvalues of this system in terms of the turbulence function alpha and its derivative alpha. They allow us to formulate an anti-dynamo theorem and a non-oscillation theorem. The conditions of these theorems, which again involve alpha and alpha, must be violated in order to reach supercritical or oscillatory regimes.
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A new class of semi-analytically solvable MHD alpha^2-dynamos is found based on a global diagonalization of the matrix part of the dynamo differential operator. Close parallels to SUSY QM are used to relate these models to the Dirac equation and to extract non-numerical information about the dynamo spectrum.
In some previous works, the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph was established. However, a specific local coordinate chart on the Grassmanian of all possible vertex conditions was used, thus creating an erroneous impression that something ``wrong can happen at the boundaries of the chart. Here we show that the analyticity of the corresponding ``dispersion relation holds over the whole Grassmannian, as well as over other parameter spaces. We also address the Dirichlet-to-Neumann (DtN) technique of relating quantum and discrete graph operators, which allows one to transfer some results from the discrete to the quantum graph case, but which has issues at the Dirichlet spectrum. We conclude that this difficulty, as in the first part of the paper, stems from the use of specific coordinates in a Grassmannian and show how to avoid it to extend some of the consequent results to the general situation.
In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the $d$-dimensional body-centered cubic (BCC) lattice $mathbb{L}^d$ and the set of non-negative integers $mathbb{Z}_+$. Thanks to the nice structure of the BCC lattice, we prove that the infrared bound holds on $mathbb{L}^dtimesmathbb{Z}_+$ in all dimensions $dgeq 9$. As opposed to ordinary percolation, we have to deal with the complex numbers due to asymmetry induced by time-orientation, which makes it hard to estimate the bootstrapping functions in the lace-expansion analysis from above. By investigating the Fourier-Laplace transform of the random-walk Green function and the two-point function, we drive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yangs bound.
We construct spectral zeta functions for the Dirac operator on metric graphs. We start with the case of a rose graph, a graph with a single vertex where every edge is a loop. The technique is then developed to cover any finite graph with general energy independent matching conditions at the vertices. The regularized spectral determinant of the Dirac operator is also obtained as the derivative of the zeta function at a special value. In each case the zeta function is formulated using a contour integral method, which extends results obtained for Laplace and Schrodinger operators on graphs.
This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set whose boundary carries Dirichlet conditions. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal-Bargmann and Hardy spaces associated with the magnetic field.
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