Eigenvalues perturbed

Akian, M., Bapat, R., and Gaubert, S. (2004). Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem, arXiv e-print math.SP/0402090. 

We are looking for the eigenvalue perturbations of the eigenfunctions which follow from the lowest eigenfimctions of the series obtained in (43) (with fc = A " = 1). We immediately use the knowledge obtained previously/ that for arbitrary weak coupling the functions combine into a symmetric and an anti-symmetric combination. The perturbation of these will be continued to... 

For equal speed of rotation (and only in this case) a passes the zero point and becomes positive, and z and 4 coincide. The eigenvalue perturbation becomes negative. It remains, however, limited, just as in the cases discussed before, and is of the order of magnitude - 35. 

Nevertheless, equation (A 1.1.145) fonns the basis for the approximate diagonalization procedure provided by perturbation theory. To proceed, the exact ground-state eigenvalue and correspondmg eigenvector are written as the sums... 

Since shallow-level impurities have energy eigenvalues very near Arose of tire perfect crystal, tliey can be described using a perturbative approach first developed in tire 1950s and known as effective mass theoiy (EMT). The idea is to approximate tire band nearest to tire shallow level by a parabola, tire curvature of which is characterized by an effective mass parameter m. ... 

In the perturbed case, assume now a small change in 7, which induces a small intersection of Bi and B2. Let B = Bi U B2 remain to be invariant. Then, we have a unique eigenmeasure /2 with eigenvalue Ai = 1 and another eigenvector P associated with A2 1. Under some continuity assumption A2 should be close to 1 and thus we have f (B) = 0. In view of (6), continuity for 7 = 7o then requires a = 1/2. Therefore (5) implies that... 

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. 

The mathematical machinery needed to compute the rates of transitions among molecular states induced by such a time-dependent perturbation is contained in time-dependent perturbation theory (TDPT). The development of this theory proceeds as follows. One first assumes that one has in-hand all of the eigenfunctions k and eigenvalues Ek that characterize the Hamiltonian H of the molecule in the absence of the external perturbation ... 

Note, that the eigenfunctions of H are /n with eigenvalues En. Compare this "exact" value with that obtained by perturbation theory in part a. 

Lowdin, P-O., Studies in Perturbation Theory. X. Bounds to Energy Eigenvalues in Perturbation Theory Ground State, Physical Review, 1965 139A 357-364. 

It should be observed that the subscript exact here refers to the lowest eigenvalue of the unrelativistic Hamiltonian the energy is here expressed in the unit Aci 00(l+m/Mz) 1 and Z is the atomic number. If the HE energies are taken from Green et al.,8 we get the correlation energies listed in the first column of Table I expressed in electron volts. The slow variation of this quantity is noticeable and may only partly be understood by means of perturbation theory. 

This is an eigenvalue problem of the form of Eq. III.45 referring to the truncated basis only, and the influence of the remainder set is seen by the additional term in the energy matrix. The relation III.48 corresponds to a solution of the secular equation by means of a modified perturbation theory,19 and the problem is complicated by the fact that the extra term in Eq. III.48 contains the energy parameter E, which leads to an iteration procedure. So far no one has investigated the remainder problem in detail, but Eq. III.48 certainly provides a good starting point. 

Expanding the sample size to 2Xc admits the other shape families shown on Fig. 6 into the analysis and leads to additional codimension-two interactions between the shapes is the (1A<.)- family and shapes with other numbers of cells in the sample. The bifurcation diagram computed for this sample size with System I and k = 0.865 is shown as Fig. 11. The (lAc)- and (Ac/2)-families are exactly as computed in the smaller sample size, but the stability of the cell shapes is altered by perturbations that are admissible is the larger sample. The secondary bifurcation between the (lAc)- and (2Ae/3)-families is also a result of a codimension two interaction of these families at a slightly different wavelength. Two other secondary bifurcation points are located along the (lAc)-family and may be intersections with the (4Ac and (4A<./7) families, as is expected because of the nearly multiple eigenvalues for these families. 

From here, the goal consists to find the eigenvalues and the eigenvectors of the perturbed system, which we denote as the sets (E,) and i> respectively. That is, the target is focused into solving the eigenvalue problem ... 

First-order perturbation theory yields the eigenvalues (diagonal elements of the permrbation matrix added to the eigenvalues of the major interaction)... 

The operator k is called the perturbation and is small. Thus, the operator k differs only slightly from and the eigenfunctions and eigenvalues of k do not differ greatly from those of the unperturbed Hamiltonian operator k The parameter X is introduced to facilitate the comparison of the orders of magnitude of various terms. In the limit A 0, the perturbed system reduces to the unperturbed system. For many systems there are no terms in the perturbed Hamiltonian operator higher than k and for convenience the parameter A in equations (9.16) and (9.17) may then be set equal to unity. 

The mathematical procedure that we present here for solving equation (9.15) is known as Rayleigh-Schrodinger perturbation theory. There are other procedures, but they are seldom used. In the Rayleigh-Schrodinger method, the eigenfunctions tpn and the eigenvalues E are expanded as power series in A... 

The non-degenerate eigenvalue E for the perturbed system to second order is obtained by substituting equations (9.24) and (9.34) into (9.20) to give... 

The Hamiltonian operator for the unperturbed harmonic oscillator is given by equation (4.12) and its eigenvalues and eigenfunctions are shown in equations (4.30) and (4.41). The perturbation H is... 

To find the perturbation corrections to the eigenvalues and eigenfunctions, we require the matrix elements for the unperturbed harmonic... 

Thus, the first-order perturbation to the eigenvalue is zero. The second-order term E is evaluated using equations (9.34), (9.50), and (4.50), giving the result... 

The perturbation method presented in Section 9.3 applies only to non-degenerate eigenvalues eX of the unperturbed system. When E is degenerate, the denominators vanish for those terms in equations (9.40) and (9.41) in which Ef is equal to E making the perturbations to E and ipn indeterminate. In... 

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