Direct eigenvalue problem

The basic scheme of this algorithm is similar to cell-to-cell mapping techniques [14] but differs substantially In one important aspect If applied to larger problems, a direct cell-to-cell approach quickly leads to tremendous computational effort. Only a proper exploitation of the multi-level structure of the subdivision algorithm (also for the eigenvalue problem) may allow for application to molecules of real chemical interest. But even this more sophisticated approach suffers from combinatorial explosion already for moderate size molecules. In a next stage of development [19] this restriction will be circumvented using certain hybrid Monte-Carlo methods. 

In the case of the second boundary-value problem with dv/dn = 0, the boundary condition of second-order approximation is imposed on 7, as a first preliminary step. It is not difficult to verify directly that the difference eigenvalue problem of second-order approximation with the second kind boundary conditions is completely posed by... 

It should be noted, however, that the Q factors of open microcavities do not characterise directly the threshold gain values of the corresponding semiconductor lasers. To overcome this difficulty a new lasing eigenvalue problem (LEP) was introduced recently (Smotrova, 2004). The LEP enables one to quantify accurately the lasing frequencies, thresholds, and near- and far-field patterns separately for various WG modes in semiconductor laser resonators. However, the threshold of a lasing mode depends on other... 

Given a state at a point (e.g., stress or strain rate) that can be described by a symmetric tensor in some orthogonal coordinate system, it is always possible to represent that particular state in a rotated coordinate system for which the tensor has purely diagonal components. The axes for such a rotated coordinate system are called the principal axes, and the diagonal components are called the principal components. Finding the principal states and the principal directions is an eigenvalue problem. 

An equivalent procedure can be used for chains containing centers with S> 1. Since the matrix dimension of the eigenvalue problem increases significantly with higher S, smaller chain segments are used to extrapolate to N —> oo, increasing the uncertainty of this approach. Therefore, for chains with S > 5/2 centers, it is better to use an expression derived for classical spins (i.e., spin vectors that are not quantized with respect to spatial directions) ... 

The eigenvalue problem of the Hamiltonian operator (1) is defined in an infinite-dimensional Hilbert space Q and may be solved directly only for very few simple models. In order to find its bound-state solutions with energies not too distant from the ground-state it is reduced to the corresponding eigenvalue problem of a matrix representing H in a properly constructed finite-dimensional model space, a subspace of Q. Usually the model space is chosen to be spanned by TV-electron antisymmetrized and spin-adapted products of orthonormal spinorbitals. In such a case it is known as the full configuration interaction (FCI) space [8, 15]. The model space Hk N, K, S, M) may be defined as the antisymmetric part of the TV-fold tensorial product of a one-electron space... 

The sign of the separation constant X2 is selected such that the homogeneous y direction results in an eigenvalue problem. In other... 

It is evident that if we are interested in physical resonances associated with complex eigenvalues of the transformed Hamiltonian H — UHU we have to avoid the trivial choices of bases. It follows directly from the so-called parent relations, Eq. (2.49), that if we like to relate the eigenvalue problem of... 

Comparing the two ways considered above one may conclude that the later approach - the generalized collective mode approach - has some important advantages. In particular, this method is especially promising in combination with molecular dynamics, because the time correlation time To (/> ), appearing in T(fc), can be directly calculated in MD simulations. Moreover, the eigenvalues problem can be formulated for initial set of nonorthogonal dynamic variables Pfc = Ak, iL- Ak, , the dynamics of... 

Zeros in the first column of H are a consequence of the fact that the cluster operator T satisfies the XCC system of equations, Eq. (78). Their presence allows us to solve the EOMXCC eigenvalue problem in the space spanned by excited configurations only and obtain energy differences ujk directly. 

In fact, it is possible to verify this directly by doing the change of variables and then applying As, and vice-versa. But this notation makes it very clear that the operator As commutes with the entire family of operators p(g), for all g. And this makes our eigenvalue problem a perfect candidate for Schur s Lemma. 

It should be observed, however, that in the literature many authors prefer to solve the eigenvalue problem (1.22) for the Liouvillian L directly in the operator space without any reference to the Hamiltonian formalism, to wave functions or ket-bra operators, and—in such a case—the algebraic conditions (1.32) are not necessarily satisfied for the approximate excitation operators D derived, and the study of these or similar conditions may become a special and sometimes crucial problem in the direct approach. ... 

At first sight, this result may seem rather uninteresting, since one of the purposes of the Liouvillian formalism is to try to solve the eigenvalue problem (1.22) directly in the operator space. On the other hand, it may be of some value if the approximate eigenoperator D = Bd obtained by solving (2.15) and (2.18) does not automatically satisfy the algebraic relations (1.32). In such a case, one may proceed by introducing an arbitrary normalized reference function associated with the reference operator... 

The key problem in the Liouvillian formalism is not only the direct solution of the eigenvalue problem (1.22) in the operator space but also— in the case of degenerate eigenvalues v—the separation of the eigenele-ments into components having the form of excitation operators of the special type (1.23) associated with specific initial and final states. Letting the superoperator G work on (1.22), one obtains... 

Still the purpose of this article is to advocate that the time is now ripe to attack the Liouvillian eigenvalue problem LC = vC directly in terms of single-commutator methods and secular equations of the type (2.16). This approach should further be combined with ket-bra methods of the type developed in Section II in order to decompose the eigenelements C associated with degenerate eigenvalues v into components having the form of excitation operators of the type C = TfXTj. ... 

The spherical harmonic analysis so far presented for uniaxial anisotropy is mainly concerned with the relaxation in a direction parallel to the easy axis of the uniaxial anisotropy. We have not considered in detail the behavior resulting from the transverse application of an external field and the relaxation in that direction for uniaxial anisotropy. Thus we have only considered potentials of the form V(r, t) = V(i, t) where the azimuthal or dependence in Brown s equation is irrelevant to the calculation of the relaxation times. This has simplified the reduction of that equation to a set of differential-difference equations. In this section we consider the reduction when the azimuthal dependence is included. This is of importance in the transition of the system from magnetic relaxation to ferromagnetic resonance. The original study [17] was made using the method of separation of variables on Brown s equation which reduced the solution to an eigenvalue problem. We reconsider the solution by casting... 

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