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Self-Adjoint Extensions of Dirac Operator with Coulomb Potential
In this note we give a concise review of the present state-of-art for the
problem of self-adjoint realisations for the Dirac operator with a Coulomb-like
singular scalar potential . We try to follow the
historical and conceptual path that leads to the present understanding of the
problem and to highlight the techniques employed and the main ideas. In the
final part we outline a few major open questions that concern the topical
problem of the multiplicity of self-adjoint realisations of the model, and
which are worth addressing in the future.Comment: 17 page
Finitely ramified iterated extensions
Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of
degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold
iterate of f, is absolutely irreducible over F; we compute a recursion for its
discriminant. Let L=L(f) be the field obtained by adjoining to F all roots, in
a fixed algebraic closure, of P_n for all n; its Galois group Gal(L/F) is the
iterated monodromy group of f. The iterated extension L/F is finitely ramified
if and only if f is post-critically finite (pcf). We show that, moreover, for
pcf polynomials f, every specialization of L/F at t=t_0 in K is finitely
ramified over K, pointing to the possibility of studying Galois groups with
restricted ramification via tree representations associated to iterated
monodromy groups of pcf polynomials. We discuss the wildness of ramification in
some of these representations, describe prime decomposition in terms of certain
finite graphs, and also give some examples of monogene number fields.Comment: 19 page
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