Schrodinger adjoint equation

Now it is well-known from the theory of self-adjoint differential equations that the lowest solution of a Schrodinger-like equation has no nodes and so it is quite impossible that the solution we seek will be identical to the (ra + l)th solution of the HF equation. 

The above problem is very much the same as the one occurring, e.g., in quantum mechanics where it is solved by the Rayleigh-Schrodinger perturbation theory. The only difference is that M is not i times a self adjoint (Hermitian) operator. For this reason, it is necessary to consider, in addition to (la), the adjoint equation... 

It follows from the definition (4.16) that, for any real value of the parameter a, the operator t is self-adjoint and the matrix (4,32) is hermitean. It is evident that the matrix < ] 11 f> > in (4.32) for z = E in some way must be similar and sometimes identical to the Bloch matrix IH I <( p>, which is easily verified by considering the power series expansions in V. It is then also clear that, as an alternative to the nonlinear Schrodinger equation H0 = 0H0, one may use multi-dimensional partitioning technique, which treats the eigenvalue problem firom a rather different point of view. It is also applicable to the case when -instead of an exact degeneracy of order p - one has p close-lying unperturbed reference states It should also be remembered that,... 

---