John V. Leahy

Degrees of controllability for quantum systems and application to atomic systems

Precise definitions for different degrees of controllability for quantum systems are given, and necessary and sufficient conditions for each type of controllability are discussed. The results are applied to determine the degree of controllability for various atomic systems with degenerate energy levels and transition frequencies.

Orbits of quantum states and geometry of Bloch vectors for N-level systems

Physical constraints such as positivity endow the set of quantum states with a rich geometry if the system dimension is greater than 2. To shed some light on the complicated structure of the set of quantum states, we consider a stratification with strata given by unitary orbit manifolds, which can be identified with flag manifolds. The results are applied to study the geometry of the coherence vector for n-level quantum systems. It is shown that the unitary orbits can be naturally identified with spheres in only for n = 2. In higher dimensions the coherence vector only defines a non-surjective embedding into a closed ball. A detailed analysis of the three-level case is presented. Finally, a refined stratification in terms of symplectic orbits is considered.

Criteria for reachability of quantum states

We address the question of which quantum states can be inter-converted under the action of a time-dependent Hamiltonian. In particular, we consider the problem as applied to mixed states, and investigate the difference between pure- and mixed-state controllabilities introduced in previous work. We provide a complete characterization of the eigenvalue spectrum for which the state is controllable under the action of the symplectic group. We also address the problem of which states can be prepared if the dynamical Lie group is not sufficiently large to allow the system to be controllable.

Eigenvalues of the form valued Laplacian for Riemannian submersions

Let π : Z → Y be a Riemannian submersion of closed manifolds. Let Φp be an eigen p-form of the Laplacian on Y with eigenvalue λ which pulls back to an eigen p-form of the Laplacian on Z with eigenvalue μ. We are interested in when the eigenvalue can change. We show that λ ≤ μ, so the eigenvalue can only increase; and we give some examples where λ < μ, so the eigenvalue changes. If the horizontal distribution is integrable and if Y is simply connected, then λ = μ, so the eigenvalue does not change. If M is a closed Riemannian manifold, let E(λ,∆pM ) be the eigenspace of the Laplacian ∆pM := dδ + δd for the eigenvalue λ on the space of smooth p-forms C∞(ΛpM). Let π : Z → Y be a Riemannian submersion of closed manifolds. Pullback defines a natural map π∗ from C∞(ΛpY ) to C∞(ΛpZ). We are interested in examples where an eigenform on Y pulls back to an eigenform on Z with a different eigenvalue. Let V and H be the vertical and horizontal distributions of π. We say that H is an integrable SL (Special Linear) distribution if H is integrable and if there exists a measure ν on the fibers of π so that the Lie derivative LHν vanishes for all horizontal lifts H ; this means that the transition functions of the fibration can be chosen to have Jacobian determinant 1. Theorem 1. Let π : Z → Y be a Riemannian submersion of closed manifolds. Let Φp ∈ E(λ,∆pY ) and π∗Φp ∈ E(μ,∆pZ). If H is an integrable SL distribution, λ = μ. We will show in Lemma 6 that if Y is simply connected and if H is integrable, then H is an integrable SL distribution and therefore eigenvalues do not change. This shows the fundamental role that the curvature tensor plays in this subject. We note that the Godbillon-Vey class of the foliation H vanishes if H is an integrable SL distribution. In the general setting, we show that if eigenvalues change, they can only increase. Theorem 2. Let π : Z → Y be a Riemannian submersion of closed manifolds. Let Φp ∈ E(λ,∆pY ) and π∗Φp ∈ E(μ,∆pZ). Then λ ≤ μ. It is not difficult to use the maximum principle to show that eigenvalues cannot change if p = 0. It is not known if eigenvalues can change if p = 1. Eigenvalues can change if p ≥ 2. Received by the editors May 20, 1996. 1991 Mathematics Subject Classification. Primary 58G25.

Weakly normal varieties: The multicross singularity and some vanishing theorems on local cohomology

The foundations for this paper were developed in [5], “Seminormal rings and weakly normal varieties”, where the historical framework and fundamental properties of weakly normal varieties were presented in detail. Here we devote our attention to the study of the multicross singularity and the role of local cohomology in the theory of weakly normal varieties.

Seminormal graded rings and weakly normal projective varieties

is paper is concerned with the seminormality of reduced graded rings and the weak normality of projective varieties. One motivation for this investi- gation is the study of the procedure of blowing up a non-weakly normal variety along its conductor ideal.

The weak subintegral closure of an ideal

Abstract For an ideal I of a ring A , we introduce the notion of the weak subintegral closure of I in an extension B of A . This gives an ideal of the weak normalization of A in B and enables us to identify the homogeneous components of the weak normalization of the Rees ring A[It] in B[t] . We get a new closure operator on ideals of A by considering the weak subintegral closure in A .

The spectral geometry of the Hopf fibration

We study the spectral geometry of the Hopf fibration and determine the right invariant metrics on for which there exist eigenforms of the Laplacian on which pull back to eigenforms of the Laplacian on . We show that the pull-back of the volume form on can be an eigenform of the Laplacian on with non-zero eigenvalue. We show that if is a principal bundle with a bundle metric and that if , then eigenvalues cannot change. Thus eigenvalues do not change for the fibrations and if . We also study the corresponding questions in the complex category for the fibration of the Hopf manifold .

Eigenforms of the spin Laplacian and projectable spinors for principal bundles

Abstract Let π : P → Y be a principal bundle with compact connected structure group G over a compact spin manifold Y. We use a suitably chosen invariant spinor on G to define pull-back operator from the spin bundle on Y to the spin bundle on P and study when the pull-back of an eigenspinor on Y is an eigenspinor on P.

Poisson structures due to lie algebra representations

Using a unitary solution of the classical Yang-Baxter equation on a Lie algebraG we describe a particular way of constructing homogeneous quadratic Poisson structures on the dual of aG-moduleV and study some local features of the symplectic foliation due to the involutive distribution of the Hamiltonian vector fields. We also give some examples where the symplectic leaves are explicitly calculated.