65 research outputs found
Landau (\Gamma,\chi)-automorphic functions on \mathbb{C}^n of magnitude \nu
We investigate the spectral theory of the invariant Landau Hamiltonian
\La^\nu acting on the space of
-automotphic functions on \C^n, for given real number ,
lattice of \C^n and a map such that the
triplet satisfies a Riemann-Dirac quantization type
condition. More precisely, we show that the eigenspace
{\mathcal{E}}^\nu_{\Gamma,\chi}(\lambda)=\set{f\in
{\mathcal{F}}^\nu_{\Gamma,\chi}; \La^\nu f = \nu(2\lambda+n) f};
\lambda\in\C, is non trivial if and only if . In such
case, is a finite dimensional vector space
whose the dimension is given explicitly. We show also that the eigenspace
associated to the lowest Landau level of
\La^\nu is isomorphic to the space, {\mathcal{O}}^\nu_{\Gamma,\chi}(\C^n),
of holomorphic functions on \C^n satisfying g(z+\gamma) = \chi(\gamma)
e^{\frac \nu 2 |\gamma|^2+\nu\scal{z,\gamma}}g(z), \eqno{(*)} that we can
realize also as the null space of the differential operator
acting on
functions on \C^n satisfying .Comment: 20 pages. Minor corrections. Scheduled to appear in issue 8 (2008) of
"Journal of Mathematical Physics
On spectral analysis of a magnetic Schrodinger operator on planar mixed automorphic forms
We characterize the space of the so-called planar mixed automorphic forms of
type with respect to an equivariant pair as the image
of the usual automorphic forms by an appropriate transform and we investigate
some concrete basic spectral properties of a magnetic Schrodinger operator
acting on them. The associated polynomials constitute classes of generalized
complex polynomials of Hermite type.Comment: 10 pages. This is a substantially reorganized, revised and improved
exposition. Misprints corrected and references added. Submitte
Resonance modes in a 1D medium with two purely resistive boundaries: calculation methods, orthogonality and completeness
Studying the problem of wave propagation in media with resistive boundaries
can be made by searching for "resonance modes" or free oscillations regimes. In
the present article, a simple case is investigated, which allows one to
enlighten the respective interest of different, classical methods, some of them
being rather delicate. This case is the 1D propagation in a homogeneous medium
having two purely resistive terminations, the calculation of the Green function
being done without any approximation using three methods. The first one is the
straightforward use of the closed-form solution in the frequency domain and the
residue calculus. Then the method of separation of variables (space and time)
leads to a solution depending on the initial conditions. The question of the
orthogonality and completeness of the complex-valued resonance modes is
investigated, leading to the expression of a particular scalar product. The
last method is the expansion in biorthogonal modes in the frequency domain, the
modes having eigenfrequencies depending on the frequency. Results of the three
methods generalize or/and correct some results already existing in the
literature, and exhibit the particular difficulty of the treatment of the
constant mode
Symplectic areas, quantization, and dynamics in electromagnetic fields
A gauge invariant quantization in a closed integral form is developed over a
linear phase space endowed with an inhomogeneous Faraday electromagnetic
tensor. An analog of the Groenewold product formula (corresponding to Weyl
ordering) is obtained via a membrane magnetic area, and extended to the product
of N symbols. The problem of ordering in quantization is related to different
configurations of membranes: a choice of configuration determines a phase
factor that fixes the ordering and controls a symplectic groupoid structure on
the secondary phase space. A gauge invariant solution of the quantum evolution
problem for a charged particle in an electromagnetic field is represented in an
exact continual form and in the semiclassical approximation via the area of
dynamical membranes.Comment: 39 pages, 17 figure
Heisenberg Evolution WKB and Symplectic Area Phases
The Schrodinger and Heisenberg evolution operators are represented in quantum
phase space by their Weyl symbols. Their semiclassical approximations are
constructed in the short and long time regimes. For both evolution problems,
the WKB representation is purely geometrical: the amplitudes are functions of a
Poisson bracket and the phase is the symplectic area of a region in phase space
bounded by trajectories and chords. A unified approach to the Schrodinger and
Heisenberg semiclassical evolutions is developed by introducing an extended
phase space. In this setting Maslov's pseudodifferential operator version of
WKB analysis applies and represents these two problems via a common higher
dimensional Schrodinger evolution, but with different extended Hamiltonians.
The evolution of a Lagrangian manifold in the extended phase space, defined by
initial data, controls the phase, amplitude and caustic behavior. The
symplectic area phases arise as a solution of a boundary condition problem.
Various applications and examples are considered.Comment: 32 pages, 7 figure
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