Landau automorphic functions on C{sup n} of magnitude {nu}
- Department of Mathematics and Computer Sciences, Faculty of Sciences, Mohammed V University, P.O. Box 1014, Agdal, 10000 Rabat (Morocco)
We investigate the spectral theory of the invariant Landau Hamiltonian, L{sup {nu}}=-1/24{sigma}{sub j=1}{sup n}{partial_derivative}{sup 2}/{partial_derivative}z{sub j}{partial_derivative}z{sub j}+2{nu}{sigma}{sub j=1}{sup n}(z{sub j}{partial_derivative}/{partial_derivative}z{sub j}-z{sub j}{partial_derivative}/{partial_derivative}z{sub j})-{nu}{sup 2}|z|{sup 2}, acting on the space F{sub {gamma}}{sub ,{chi}}{sup {nu}} of ({gamma},{chi})-automorphic functions on C{sup n}, constituted of C{sup {infinity}} functions satisfying the functional equation f(z+{gamma})={chi}({gamma})e{sup i{nu}}{sup Imlangz,{gamma}}{sup rang}f(z); z(set-membership sign)C{sup n},{gamma}(set-membership sign){gamma}, for given real number {nu}>0, lattice {gamma} of C{sup n} and a map {chi}:{gamma}{yields}U(1) such that the triplet ({nu},{gamma},{chi}) satisfies a Riemann-Dirac quantization-type condition. More precisely, we show that the eigenspace E{sub {gamma}}{sub ,{chi}}{sup {nu}}({lambda})={l_brace}f(set-membership sign)F{sub {gamma}}{sub ,{chi}}{sup {nu}}; L{sup {nu}}f={nu}(2{lambda}+n)f{r_brace}; {lambda}(set-membership sign)C, is nontrivial if and only if {lambda}=l=0,1,2,...,. In such case, E{sub {gamma}}{sub ,{chi}}{sup {nu}}(l) is a finite dimensional vector space whose the dimension is given explicitly by dim E{sub {gamma}}{sub ,{chi}}{sup {nu}}(l)=(n+l-1; l;)({nu}/{pi}){sup n}vol(C{sup n}/{gamma}). Furthermore, we show that the eigenspace E{sub {gamma}}{sub ,{chi}}{sup {nu}}(0) associated with the lowest Landau level of L{sup {nu}} is isomorphic to the space, O{sub {gamma}}{sub ,{chi}}{sup {nu}}(C{sup n}), of holomorphic functions on C{sup n} satisfying g(z+{gamma})={chi}({gamma})e{sup {nu}}{sup /2|{gamma}}{sup |{sup 2}}{sup +{nu}}{sup <z,{gamma}}{sup >}g(z= ), (*) that we can realize also as the null space of the differential operator, {sigma}{sub j=1}{sup n}(-{partial_derivative}{sup 2}/{partial_derivative}z{sub j}{partial_derivative}z{sub j}+{nu}z{sub j}{partial_derivative}/{partial_derivative}z{sub j}) acting on C{sup {infinity}} functions on C{sup n} satisfying (*)
- OSTI ID:
- 21100359
- Journal Information:
- Journal of Mathematical Physics, Vol. 49, Issue 8; Other Information: DOI: 10.1063/1.2958090; (c) 2008 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
Similar Records
The holonomy expansion: Invariants and approximate supersymmetry
Eightfold way from dynamical first principles in strongly coupled lattice quantum chromodynamics