Relationship between sectional curvature and null spaces of Lichnerowicz-type Laplacians and their smallest eigenvalues

In the paper, we prove that the curvature operator of the second kind of Riemannian manifolds is positive (respectively, negative) if and only if its sectional curvature is also positive (respectively, negative). In addition, we prove several vanishing theorems on null spaces of the Lichnerowicz, Sampson and Hodge-de Rham Laplacians and we find estimates of their lowest eigenvalues on closed Riemannian manifolds of sign-definite sectional curvature

Integrability and action operators in quantum Hamiltonian systems

For a (classically) integrable quantum mechanical system with two degrees of freedom, the functional dependence $\hat{H}=H_Q(\hat{J}_1,\hat{J}_2)$ of the Hamiltonian operator on the action operators is analyzed and compared with the corresponding functional relationship $H(p_1,q_1;p_2,q_2) = H_C(J_1,J_2)$ in the classical limit of that system. The former is shown to converge toward the latter in some asymptotic regime associated with the classical limit, but the convergence is, in general, non-uniform. The existence of the function $\hat{H}=H_Q(\hat{J}_1,\hat{J}_2)$ in the integrable regime of a parametric quantum system explains empirical results for the dimensionality of manifolds in parameter space on which at least two levels are degenerate. The comparative analysis is carried out for an integrable one-parameter two-spin model. Additional results presented for the (integrable) circular billiard model illuminate the same conclusions from a different angle.Comment: 9 page

Semiclassical treatment of logarithmic perturbation theory

The explicit semiclassical treatment of logarithmic perturbation theory for the nonrelativistic bound states problem is developed. Based upon $\hbar$-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions for the one-dimensional anharmonic oscillator is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and exited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues of the harmonic oscillator perturbed by $\lambda x^{6}$ are considered.Comment: 6 pages, LATEX 2.09 using IOP style

Signatures of quantum integrability and nonintegrability in the spectral properties of finite Hamiltonian matrices

For a two-spin model which is (classically) integrable on a five-dimensional hypersurface in six-dimensional parameter space and for which level degeneracies occur exclusively (with one known exception) on four-dimensional manifolds embedded in the integrability hypersurface, we investigate the relations between symmetry, integrability, and the assignment of quantum numbers to eigenstates. We calculate quantum invariants in the form of expectation values for selected operators and monitor their dependence on the Hamiltonian parameters along loops within, without, and across the integrability hypersurface in parameter space. We find clear-cut signatures of integrability and nonintegrability in the observed traces of quantum invariants evaluated in finite-dimensional invariant Hilbert subspaces, The results support the notion that quantum integrability depends on the existence of action operators as constituent elements of the Hamiltonian.Comment: 11 page

Renormalized theory of the ion cyclotron turbulence in magnetic field--aligned plasma shear flow

The analytical treatment of nonlinear evolution of the shear-flow-modified current driven ion cyclotron instability and shear-flow-driven ion cyclotron kinetic instabilities of magnetic field--aligned plasma shear flow is presented. Analysis is performed on the base of the nonlinear dispersion equation, which accounts for a new combined effect of plasma turbulence and shear flow. It consists in turbulent scattering of ions across the shear flow with their convection by shear flow and results in enhanced nonlinear broadening of ion cyclotron resonances. This effect is found to lead to the saturation of ion cyclotron instabilities as well as to the development of nonlinear shear flow driven ion cyclotron instability. 52.35.RaComment: 21 page