Quantization of periodic motions on compact surfaces of constant negative curvature in a magnetic field

Abstract

Abstract-We use the semiclassical approach to study the spectral problem for the Schr¨odinger operator of a charged particle confined to a two-dimensional compact surface of constant negative curvature. We classify modes of classical motion in the integrable domain E < E cr and obtain a classification of semiclassical solutions as a consequence. We construct a spectral series (spectrum part approximated by semiclassical eigenvalues) corresponding to energies not exceeding the threshold value E cr ; the degeneration multiplicity is computed for each eigenvalue.

Trace formula for the magnetic Laplacian

“…It is well known that, if F satisfies the quantization condition (6), then such a Hermitian line bundle (L, h L ) with Hermitian connection ∇ L exists. The Riemannian metric on X and the Hermitian structure on L allows us to define inner products on C ∞ (M, L) and C ∞ (M, T * M ⊗ L) and the adjoint operator…”

“…If the set of periods of the flow is discrete, cleanness of the flow is equivalent to the following condition: for every period T , the fixed point set of φ T is a submanifold of X E , and at each x ∈ P T the tangent space of P T coincides with the set of fixed vectors of the tangent map d(φ T ) x . Now suppose that the magnetic field form F satisfies the condition (6). Recall the notion of action for closed curves in T * M .…”

Trace formula for the magnetic Laplacian on a compact hyperbolic surface

“…Indeed, for every point q = (p.ξ) ∈ SM we consider the hyperbolic circle γ on M such that γ(0) = p, γ(0) = ξ, take the center c γ of this circle and put F (q) = f (c γ ). These integrals were successfully used in [2] for a quantization of periodic magnetic geodesics.…”

Quasiclassical approximation for magnetic monopoles

“…We remark that a construction of a canonical operator on an arbitrary symplectic manifold was proposed in [5]. However such an operator takes values in a certain bundle of wave packets in difference with (2) where the canonical operator takes values in the same space as the approximated operator. In [2], the operator from [5] was used for constructing spectral series for the quantum Hamiltonian of a charged particle in constant nonzero magnetic field on a hyperbolic surface.…”

“…However such an operator takes values in a certain bundle of wave packets in difference with (2) where the canonical operator takes values in the same space as the approximated operator. In [2], the operator from [5] was used for constructing spectral series for the quantum Hamiltonian of a charged particle in constant nonzero magnetic field on a hyperbolic surface. In this case, the magnetic field becomes exact on the universal covering of the surface, the particle trajectories are geodesic circles, any smooth function on the surface defines a first integral of the flow [16] -a closed trajectory corresponds to the value of the function in the center of the geodesic circle, and, starting from any such smooth function, one can construct a quasiclassical approximation to describe the series of eigenvalues of the magnetic Laplacian below some threshold level.…”