Bivariational principle

In the same way that a self-adjoint operator satisfies a variational principle, a pair of adjoint operators T and Tf satisfy a bivariational principle (5), which may also be applied to the transformed operators f and rT. Let us start by considering the variational expressions... 

One may use the bivariational principle to calculate approximate eigenfunctions and eigenvalues to the operators T and T. For this purpose, we will introduce two linearly independent sets of order m... 

Using the bivariational principle to determine the column vectors c and d, one gets immediately the linear relations... 

It is clear that the concept of outer projections permits us to formulate the approximate treatment of the eigenfunctions and eigenvalues based on the bivariational principle in an exact manner. 

By using the bivariational principle and proper choice of bases, it is hence possible to transfer some of the characteristic properties of the exact eigenfunctions to the approximate ones. 

It should be observed that, in the approximate treatment of the eigenvalue problem based on the bivariational principle and the use of a truncated basis

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Even in the approximate treatment using the bivariational principle and a truncated finite basis, it is usually easily checked that the eigenfunctions f a corresppnding to the persistent eigenvalues belong to the domain D(u 1) of... 

Fig. 2. The change of the energy spectrum under complex scaling. (A) The spectrum of the original Hamiltonian (B) the exact spectrum of the dilated Hamiltonian (C) the approximate spectrum with all the eigenvalues discrete as obtained by the bivariational principle and a truncated finite basis.

This relation is sometimes referred to as the bivariational principle,5 and it plays an important role in the current literature. 

The last member of (3.31) is a typical bivariational expression, but—since the Liouvillian L is self-adjoint—one should expect that the bivariational principle (3.15) should lead to a solution, where Ct is proportional to C. This is usually not the case, unless some specific conditions are satisfied, and one is hence facing about the same problems as discussed before in connection with the quantity l, and one may then resort to the methods developed in Section II. 

In this chapter, I will try to explain some of the statistical methods (univariate, bivariate and multivariate) most used by our group, in the hope that it will be useful and accessible for the majority of readers, given that the emphasis is on comprehension of the principles of the methods, their applications and interpretation of the results obtained. 

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