Adjoint eigenfunctions

Ao is the eigenvalue whose existence is established by means of the preceding lemmas. Its simplicity can be established by applying Theorem 1 to Equation (14) with a = Ao. Moreover, by construction, the corresponding eigenfunction is quasi-positive and the adjoint eigenfunction is positive. The dominance of Ao can be shown by use of the same technique as applied in proving Theorem 1. 

Berezanskij, Yu. (1968) Expansion,s in Eigenfunctions of Self-Adjoint Operator. AMS Providence, RI. 

For an isolated system, H(x) is time independent, eq. (3) is separated as usual leading to the time independent equation H(x) < x d)> = E < x d>>. The structure of H is not known in detail. So far, it is just a symbol, but if this is a hermitian and self-adjoint operator, there exists a complete denumerable set of eigenfunctions. 

The stochastic resonance is determined by the longitudinal (with respect to n) modes of the relaxational problem (4.90). Since A is not a self-adjoint operator, it produces, together with the spectrum of eigenvalues ,, two sets of eigenfunctions defined as... 

We shall return to the derivation of the individual terms of the effective Hamiltonian listed in (7.43) for some particular cases later in this chapter. To conclude this section, we now consider some of the general properties of the operator Xeff (0) and its eigenfunctions. Each of the terms listed in (7.43) is composed of products of the three operators Pq, Qo and X7. Although each of these operators is individually Hermitian (i.e. the operator is self-adjoint, pj, = P0), a product of any of them is not necessarily Hermitian. In fact, a term is only Hermitian when it is palindromic, that is, when it reads the same forwards as backwards. Inspection of equation (7.43) reveals that 3Ceff(0) is Hermitian up to and including terms in X2 but that there are non-Hermitian terms in the /.3 and higher-order contributions. The nature ofthese non-Hermitian properties can... 

Special Case when T = T = T. Let us now consider the special case when a complex symmetric operator is real, so that T = T. In this case, the operator T is also self-adjoint, T = T, and one can use the results of the conventional Hartree-Fock method 7. The eigenvalues are real, X = X. and - if an eigenvalue X is non-degenerate, the associated eigenfunction C is necessarily real or a real function multiplied by a constant phase factor exp(i a). In both cases, one has D = C 1 = C. In the conventional Hartree-Fock theory, the one-particle projector p takes the form... 

Since L is self-adjoint, the eigenfunctions y, and y, corresponding to different eigenvalues and Aj, respectively, are orthogonal, i.e.. 

The Usachev Gandini derivation of GPT is based on physical considerations. Their formulation is applicable to alterations that leave the reactor critical. Stacey (40) derived GPT from variational principles. His GPT formulations are also applicable to perturbations that change the static eigenvalue of the Boltzmann equation that is, that do not preserve criticality. The approach used in this work for deriving GPT expressions is neither that of the variational, nor of the physical consideration. It uses conventional perturbation techniques combined with the flux-difference and adjoint-difference methods (see Section III,B). A third version of GPT is presented in this work. Like Stacey s this new version is applicable to perturbations that do not preserve criticality. It pertains, however, to integral parameters that are related to the prompt-mode rather than to the static eigenfunctions. At the end of this section we discuss the relation between... 

Having defined the eigenfunctions (i.e., the orthogonal basis), the inner product and the self-adjoint property, we are ready to apply the integral transform to the physical system (Eq. 11.189). 

Also the eigenfunctions are not orthogonal to each other, but to a set of adjoint functions v>, satisfying the adjoint equation E — = 0. 

The combination of transposition and complex conjugation is called the adjoint operation, indicated by a dagger. A Hermitian matrix is thus self-adjoint. An eigenfunction of this matrix, operating in a function space, may be expressed as a linear combination... 

If Ho is self-adjoint then the eigenfunctions of the zero-order problem are orthonormal... 

Theorem 2 shows that /x2, and hence is a continuous function of a for a > 0. Let O be the quasi-positive eigenfunction corresponding to the dominant eigenvalue of Equation (14) and be the positive adjoint... 

In (2.16) the functions ifjj are, of course, the eigenfunctions corresponding to the eigenvalues Aj. It can be shown that the adjoint operator A has the same eigenvalues Xj, Its eigenfunctions, ifff, are different from the ifjj. 

Equation (29) is now solved for p by a perturbation comparison with Equation (30). Let n, 7 o represent the eigenfunctions of these equations, and m, rriQ represent the eigenfunctions of the adjoint equations... 

Define an iterative series of adjoint functions (not eigenfunctions) by the algorithms... 

---