Eigenvalue problem, reduced

If the functions Oj are orthonormal, then the overlap matrix S reduces to the unit matrix and the above generalized eigenvalue problem reduces to the more familiar form ... 

The simplest truncation of the eigenvalue equation (2) for the excitation energies is to ignore all coupling between poles, except that between a singlet-triplet pair. This is equivalent to setting (gl/nxcl ) to zero, for q q. (We have dropped the spin-index on these contributions, since we deal only with closed shell systems). Then the eigenvalue problem reduces to a simple 2x2 problem, with solutions... 

This is an important general result which relates the free energy per particle to the largest eigenvalue of the transfer matrix, and the problem reduces to detennining this eigenvalue. 

The variational Ritz procedure reduces the problem of solving (1) to solving the generalized eigenvalue problem ... 

The single sum in Eq. III.99 is here replaced by the sum of m such single sums, depending on the orthogonalization. The eigenvalue problem (Eq. III.21) is in this way reduced to the form... 

These coefficients (Equation 7.30) are required to calculate the transition probability or spectral amplitude (cf. Chapter 8). Note that for systems with more than two spin wavefunctions (S > 1/2) the energy eigenvalue problem is usually not solvable analytically (unless the matrix can be reduced to one of lower dimensionality because it has sufficient off-diagonal elements equal to zero) and numerical diagonalization is the only option. 

Given ng and 1(R, p), Eqs. (7.32a, b) can be integrated successively from r = R to a large value of r. By definition Z(R, p) = -y where yis the recombination probability in presence of scavenger. Only for the correct value of ydo the solutions of (7.32a, b) smoothly vanish asymptotically as r—-o° otherwise, they diverge. Thus, the mathematics is reduced to a numerical eigenvalue problem of finding the correct value of I(R, p). 

Thus, with a nearly zero eigenvalue of the covariance matrix of the independent variables the estimates tend to be inflated and the results are meaningless. Therefore, in nearly singular estimation problems reducing the mean square... 

One of the most important concepts of quantum chemistry is the Slater determinant. Most quantum chemical treatments are made just over Slater determinants. Nevertheless, in many problems the formulation over Slater determinants is not very convenient and the derivation of final expressions is very complicated. The advantage of second quantization lies in the fact that this technique permits us to arrive at the same expressions in a considerably simpler way. In second quantization a Slater determinant is represented by a product of creation and annihilation operators. As will be shown below, the Hamiltonian can also be expressed by creation and annihilation operators and thus the eigenvalue problem is reduced to the manipulation of creation and annihilation operators. This manipulation can be done diagrammatically (according to certain rules which will be specified later) and from the diagrams formed one can write down the final mathematical expression. In the traditional way a Slater determinant I ) is specified by one-electron functions as follows ... 

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... 

So, to use the spin permutation technique we constructed the symmetry adapted lattice Hamiltonian in a compact operator form and essentially reduced the dimensionality of the corresponding eigenvalue problem. The effects of tpp 0 and the additional superexchange of copper holes are considered in [48]. 

For co = 0, this equation reduces to (14), whereas dm/dz = 0 reproduces (23). Mathematically, Eq. (24) is a well-known random-potential eigenvalue problem, which can be solved numerically or by transfer-matrix methods [5, 153],... 

One approach to calculating the stationary mutant distributions for longer sequences is to form classes of sequences within the quasi-species. These classes are defined by means of the Hamming distance between the master sequence and the sequence under consideration. Class 0 contains the master sequence exclusively, class 1 the v different one-error mutants, class 2 all v(v —1)/2 two-error mutants, and so on. In general we have all (JJ) fe-error mutants in class k. In order to be able to reduce the 2 -dimensional eigenvalue problem to dimension v 1, we make the assumption that all formation rate constants are equal within a given class. We write Aq for the master sequence in class 0, Ai for all one-error mutants in class 1, 4 2 for all two-error mutants in class 2, and in general A for all k error mutants in class k. 

It may be shown5 that one may solve the eigenvalue problem for H exactly within the subspace of Q, if and only if the projector Q reduces H, so that... 

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... 

Thus, for two free surfaces, the eigenvalue problem for a is reduced to finding a nontrival solution of Eq. (12-199), subject to the six boundary conditions, (12-200), (12-203), and (12-204). In particular, let us suppose that we specify Pr, Gr, and a2 (the wave number of the normal model of perturbation). There is then a single eigenvalue for a such that / / 0. If Real(er) < 0 for all a, the system is stable to infinitesimal disturbances. On the other hand, if Real(er) > 0 for any a, it is unconditionally unstable. Stated in another way, the preceding statements imply that for any Pr there will be a certain value of Gr such that all disturbances of any a decay. The largest such value of Gr is called the critical value for linear stability. 

Now the problem is reduced to an eigenvalue problem for the matrix H-A that is, one must evaluate the eigenvalues Xp of linear equations... 

---