Scattering of Dirac particles by mabnetic fields and spectral theory

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The subject of this research project is the spectral theory for Dirac operators with magnetic fields in two dimensions, and the special emphasis is placed on the study about the relation between scattering of Dirac particles by magnetic fields and resonance at zero energy of Schrodinger operators. The motion of massless Dirac particles is governed by the operator D(A,V)=σ・ (-i∇-A)+V acting on [L^2(R^2)]^2, where A(x):R^2→R^2 is a magnetic potential, V(x):R^2→R is a electronic potential and σ=(σ_1,σ_2) is a vector with 2 × 2 Pauli matrices as components. The square of Dirac Operator D(A,0) without scalar potential V becomes the diagonal operator with Schrodinger operators H_±=(-i∇-A)^2±b as diagonal components (Pauli operator), where b=∇×A:R^2→R denotes the magnetic field associated with vector potential A. The both operators H_± 【greater than or equal】 0 are nonnegative, but they have a different spectral structure at zero energy. If, for example, b∈C^∞_0(R^2) is compactly supported an … More d it has the noninteger flux α defined by α=∫b(x)dx/2π 【not a member of】 Z, then equation H_u=0 has a bounded solution (resonance) not in L^2, while H_+u=0 does not have such a solution. Thus the zero energy resonance of Schrodinger operators appears in the spectral theory for Dirac operators in a quite natural way. The present project deals with the following two subjects closely related to zero energy resonance : (1)resolvent convergence in norm to Dirac operators with solenoidal magnetic fields (point-like fields) : (2)scattreing by electromagnetic fields with small support. These subjects have both been discussed in physical articles when electromagnetic fields are spherically symmetric, and the results obtained are based on a calculation using the Bessel functions. The main achievement is that we have made clear the role of zero energy resonance hidden behind the explicit calculation from a mathematical point of view by eliminating the assumption of spherical symmetry. The scattering of Dirac particles by electromegnetic fields with small support appears in the model of cosmic string as an important problem of mathematical physics. The application to it has been also studied. Less