Orthogonal eigenfunctions

Since we shall usually be more interested in the eigenvalues than the eigenfunctions ofXett (0), the possible non-Hermitian properties need not concern us too deeply. However, there may be certain situations, for example in the use of computer programmes to extract the eigenvalues of Xeff (0), where we must take account of the non-orthogonal eigenfunctions. In these situations, it is easiest to use a symmetrised form of the non-Hermitian terms in Xeff (0) in the third order, this would be... 

EOF analysis in the context of turbulent flows was introduced by Lumley [381], It consists of finding the sequence of orthogonal eigenfunctions and associated eigenvalues that converges optimally fast when the variance of the turbulent field is represented... 

Hence there is no need to put j as an additional label to the eigenfunctions For each set of eigenvalues ruj and Kj one finds indeed two orthogonal eigenfunctions hence the corresponding common eigenspace... 

The partial wave Dirac operator is independent of the quantum number ruj. Hence, if solve the above eigenvalue equation, then there are 2j + 1 orthogonal eigenfunctions of H that belong to the same eigenvalue E. These eigenfunctions are labelled by the quantum number mj = —j, j + 1,..., +j in... 

We will assume that both the wavefunction and the orthogonal eigenfunctions are normalized, which implies that... 

The Helmholtz operator L in (1) is well known to have a basis of real orthogonal eigenfunctions with real eigenvalues... 

To evaluate the integrals on the right side of this expression, first expand ip formally in terms of a set of normalized, orthogonal eigenfunctions of H for the system ... 

Only where are intersected the lines and the colunms from the determinant, belonging of the same irreducible representation, will be non-null elements, based on the great orthogonality theorem (2.102) which recommend SALC as being orthogonal eigenfunctions. It also takes into account the adjacency of x SALC type which prescribes unitary contributions... 

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