General Eigenfunction Expansion

We seek a solution of (7-99) by means of the method of separation of variables. For this purpose, it is more convenient to note that E4f = 0 can be split into two second-order equations, 

Obviously, substituting (7-105) into (7-104), we recover (7-99). In the present analysis, we first determine the most general solution for co by solving (7-104), and then we solve (7-105) for i with co given by this general solution. 

The first term is a function of r only, whereas the second is a function of rj. Hence it follows that each must equal a constant. We denote this constant as n(n +1), and thus (7-107) separates into two equations, 

Equation (7-108) is a particular case of Euler s equation, for which a general solution is 

Except for the special case, n = 0, a conveniant approach to solving (7-109) is to differentiate the whole equation with respect to rj because this transforms it into an equation of well-known form, namely, 

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I. General Eigenfunction Expansions in Cartesian and Cylindrical Coordinates... 

Problem 7-20. Sphere in a Parabolic Flow. Use the general eigenfunction expansion for axisymmetric creeping-flows, in spherical coordinates, to determine the velocity and pressure fields for a sohd sphere of radius a that is held fixed at the central axis of symmetry of an unbounded parabolic velocity field,... 

Although the problem defined by (3-95) and (3-96) is time dependent, it is linear in uJ and confined to the bounded spatial domain, 0 < r < 1. Thus it can be solved by the method of separation of variables. In this method we first find a set of eigensolutions that satisfy the DE (3-95) and the boundary condition at r = 1 then we determine the particular sum of those eigensolutions that also satisfies the initial condition at 7 = 0. The problem (3-95) and (3-96) comprises one example of the general class of so-called Sturm-Louiville problems for which an extensive theory is available that ensures the existence and uniqueness of solutions constructed by means of eigenfunction expansions by the method of separation of variables.14 It is assumed that the reader is familiar with the basic technique, and the solution of (3-95) and (3-96) is simply outlined without detailed proofs. We begin with the basic hypothesis that a solution of (3-95) exists in the separable form... 

We begin with a powerful solution method that can be applied for general 3D flows whenever the boundaries of the domain can be expressed as a coordinate surface for some orthogonal coordinate system. In this case, we can use an invariant vector representation of the velocity and pressure fields to simultaneously represent (solve) the solutions for a complete class of related problems by using so-called vector harmonic functions, rather than solving one specific problem at a time, as is necessary when we are using standard eigenfunction expansion techniques. 

However, this case is exceptional in the sense that for all other choices of E and pressure fields will be fully 3 D and the eigenfunction expansion of the preceding chapter cannot be used. In addition, instead of focusing on a specific example of the general class of undisturbed flows (8 36), as would be required by any of the methods of solution discussed in the previous chapter, we show that the present method allows the solution to be completed for arbitrary E and simultaneously determine the solution for all possible flows of the general class (8-36). 

In the case of the one-dimensional multigroup diffusion operator, Q, it is shown that the spectral subspaces generated by the generalized eigenfunctions of Q are complete, and biorthogonal expansions in these generalized... 

It is shown in Appendix 6 that the generalized Laguerre polynomials are eigenfunctions of the integral operator (3.26) with kernel (3.52). Let us search for the solution of (3.26) in the form of expansion over these eigenfunctions... 

Obviously, the state function (jc) is not an eigenfunction of H. Following the general procedure described above, we expand in terms of the eigenfunctions n). This expansion is the same as an expansion in a Fourier series, as described in Appendix B. As a shortcut we may use equations (A.39) and (A. 40) to obtain the identity... 

E. A. Hylleraas, Z Phys. 65 (1930), 209 note 6, p. 279. Note that if(2) can alternatively be expressed as an infinite expansion in the unperturbed eigenfunctions but the Hylleraas variation-perturbation expression (1.5d) is generally more useful for practical numerical applications. 

The situation here is completely analogous to that obtained in the restricted open HE theory (ROHF). The states are not eigenfunctions of S, except when all the open-shell electrons have parallel spins (A p = 0 or = 0). This result is a consequence of the expansion (22) used to obtain Eq. (27). Actually, the spin decomposition, Eq. (22), for does not conserve in general the total spin S. However, we can form appropriate linear combinations of two-electron spin functions cri Sj, 8 81,52) that are simultaneously eigenfunctions of and S, and achieve a correct spin decomposition of the 2-RDM [85] ... 

Friazinov (F4) deals with a generalized Stefan problem involving finite depth of the two-phase layer, densities and thermal conductivities which are functions of position, and arbitrary initial and boundary conditions, by an approximate expansion in terms of appropriate Sturm-Liouville eigenfunctions. 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

In general, an arbitrary ket is not an eigenfunction of the operator. It may be verified, however, that the three operators H, and all commute with each other. This means that it is possible to construct wavefunctions that are simultaneous eigenfunctions of all three of these operators with suitable linear combinations of expansion kets. 

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