Stable subspace

However, a Cl hamiltonian matrix have eigenvalues spread out in a very wide range, and this reduces the value of the Lanczos algorithm for Cl purposes. Attempts have been made to precondition the Lanczos procedure, such that the iterations are governed by a clustered matrix while at the same time a stable subspace of the original matrix is constructed. The success of such a scheme has yet to be demonstrated. 

If an eigenvalue X - as defined by the characteristic equation for T in any matrix representation - is degenerate, the situation is more complicated, and the eigenvalue problems (2.3) have to be replaced by the associated stability problems see ref. B, Sec. 4. In matrix theory, the search for the irreducible stable subspaces of T is reflected in the block-diagonalization of the matrix... 

We note that, due to the bi-orthogonality theorem, the metric matrix A = < C I C > is automatically block-diagonalized with different blocks for different eigenvalues, and it is then possible to treat the stable subspaces associated with different eigenvalues seperately, which is an essential simplification. In addition, the symmetry property (2.24) indicates another simplification from the computational point of view. 

It is well known that the stable manifold A (x, 0) of a hyperbolic, unstable rest point (x, 0) has Lebesgue measure zero. This follows from Sard s theorem (see Appendix E) and the fact that the stable manifold is the image of a smooth one-to-one map of into K" x K ", where /, is the dimension of the stable subspace of the linearization of (F.l) about (x, 0) and consequently li[Pg.296]

By contrast, effective Hamiltonian methods produce a stable subspace and an effective Hamiltonian matrix ... 

Instead of working in a stable subspace of (i.e. in a basis of lOp determinants), one may work in a basis of a rather limited number of configurations, those which play a major role in the adiabatic eigenfunctions of interest, and build an effective electronic Hamiltonian in this model space. For instance, for the curve crossing between the ionic and neutral configuration in NaCl, one may define as a model space the two leading configurations... 

This is exactly the autonomous linearized Hamiltonian (7), the dynamics of which was discussed in detail in Section II. One therefore finds the TS dividing surface and the full set of invariant manifolds described earlier one-dimensional stable and unstable manifolds corresponding to the dynamics of the variables A<2i and APt, respectively, and a central manifold of dimension 2N — 2 that itself decomposes into two-dimensional invariant subspaces spanned by APj and AQj. However, all these manifolds are now moving manifolds that are attached to the TS trajectory. Their actual location in phase space at any given time is obtained from their description in terms of relative coordinates by the time-dependent shift of origin, Eq. (42). 

At Da = 0.4 (Fig. 4.29(b)), the two saddle points from the pure vertices move into the composition triangle. The stable node from the 1,4-BD vertex moves to the kinetic azeotrope at x = (0.0328, 0.6935). Pure water and pure THF now become stable nodes. The unstable node between water and THF remains unmoved, and forms two separatrices with the two saddle points. Thereby, the whole composition space is divided into three subspaces which have each a stable node, namely pure water, pure THF and the kinetic azeotrope. 

The problem treated in this paper is complicated by the fact that the linear operator T is supposed to be linearly defined on a linear space A = F of all complex functions F = F(X) of a real composite variable X = jcl5 x2, , xN, of which the L2 Hilbert space is only a small subspace. We will refer to this space A = F as the definition space of the operator T. Since it contains all complex functions, it is stable under complex conjugation, and, according to Eq. (1.50) one has T F = (TF ), which means that also the complex conjugate operator T is defined on this space. 

Theorem. Let kbe a field of characteristic zero. Let G be a connected affine algebraic group schemeacting linearly on V. A subspace W of Vis stable under G iff it is stable under Lie(G). 

Appendix A. Treatment of One-Particle Subspaces of Order N which are Stable under Complex Conjugation. 

The nessaiy and sufficient condition that the subspace spanned by the set p is stable under complex conjugation is that the matrix a = oplnp> 1opl ip > satifies the relation... 

Let us now consider the dynamics of the coupled system with Hamiltonian H = Hp, + Hp. Ju and Jp remain good quantum numbers for this Hamiltonian and are quantized according to Eq. (31). It is known that the dynamics of the coupled system is governed by the shape of its stable periodic orbits (POs) in the subspace of the normal coordinates involved in the Fermi... 

The behavior of orbits of P near a fixed point x can be described in the case where x is a hyperbolic fixed point, that is, when no eigenvalue (multiplier) of the Jacobian of P at x has modulus equal to 1. In this case there exist (local) stable and unstable manifolds M (x) and M (x) (respectively) containing the point x which are tangent to the stable (resp. unstable) subspace of the Jacobian of P at x. (The stable (unstable) subspace... 

As the dimension of the blocks of the Hessian matrix increases, it becomes more efficient to solve for the wavefunction corrections using iterative methods instead of direct methods. The most useful of these methods require a series of matrix-vector products. Since a square matrix-vector product may be computed in 2N arithmetic operations (where N is the matrix dimension), an iterative solution that requires only a few of these products is more efficient than a direct solution (which requires approximately floating-point operations). The most stable of these methods expand the solution vector in a subspace of trial vectors. During each iteration of this procedure, the dimension of this subspace is increased until some measure of the error indicates that sufficient accuracy has been achieved. Such iterative methods for both linear equations and matrix eigenvalue equations have been discussed in the literature . 

Since the synchronous CW solution is stable within the synchronization subspace, its stability in the whole phase space is determined by its transverse stability, i.e. the stability with respect to perturbations of the initial data transverse to the synchronization subspace. The analysis of the transverse stability of synchronous CW solutions can be carried out by inspecting the transverse characteristic equation [29]... 

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