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Register a new userSymmetry operators and separation of variables for Diracs equation on two-dimensional spin manifolds with external fields
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The second order symmetry operators that commute with the Dirac operator with external vector, scalar and pseudo-scalar potentials are computed on a general two-dimensional spin-manifold. It is shown that the operator is defined in terms of Killing vectors, valence two Killing tensors and scalar fields defined on the background manifold. The commuting operator that arises from a non-trivial Killing tensor is determined with respect to the associated system of Liouville coordinates and compared to the the second order operator that arises from that obtained from the unique separation scheme associated with such operators. It shown by the study of several examples that the operators arising from these two approaches coincide.
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We demonstrate how the Moutard transformation of two-dimensional Schrodinger operators acts on the Faddeev eigenfunctions on the zero energy level and present some explicitly computed examples of such eigenfunctions for smooth fast decaying potentials of operators with non-trivial kernel and for deformed potentials which correspond to blowing up solutions of the Novikov-Veselov equation.
We show that all results of Yasar and Ozer [Comput. Math. Appl. 59 (2010), 3203-3210] on symmetries and conservation laws of a nonconservative Fokker-Planck equation can be easily derived from results existing in the literature by means of using the fact that this equation is reduced to the linear heat equation by a simple point transformation. Moreover nonclassical symmetries and local and potential conservation laws of the equation under consideration are exhaustively described in the same way as well as infinite series of potential symmetry algebras of arbitrary potential orders are constructed.
Hamiltonian operators are used in the theory of integrable partial differential equations to prove the existence of infinite sequences of commuting symmetries or integrals. In this paper it is illustrated the new Reduce package cde for computations on Hamiltonian operators. cde can compute the Hamiltonian properties of skew-adjointness and vanishing Schouten bracket for a differential operator, as well as the compatibility property of two Hamiltonian operators and the Lie derivative of a Hamiltonian operator with respect to a vector field. It can also make computations on (variational) multivectors, or functions on supermanifolds. This can open the way to applications in other fields of Mathematical Physics.
Using Moutard transformations we show how explicit examples of two-dimensional Schroedinger operators with fast decaying potential and multidimensional $L_2$-kernel may be constructed
We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere S^d in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x-y|^(-s) with d-2 <= s < d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on S^d is determined. The special case s = d-2 yields interesting phenomena, which we investigate in detail. A weak* asymptotic analysis is provided as s goes to (d-2)^+.
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