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The Evans Lemma is basic for Myron W. Evans' GCUFT or ECE Theory. Evans has given two proofs of his Lemma. Both proofs are shown here to be in error and beyond repair.
We propose a method for the treatment of two--point boundary value problems given by nonlinear ordinary differential equations. The approach leads to sequences of roots of Hankel determinants that converge rapidly towards the unknown parameter of the problem. We treat several problems of physical interest: the field equation determining the vortex profile in a Ginzburg--Landau effective theory, the fixed--point equation for Wilson's exact renormalization group, a suitably modified Wegner--Houghton's fixed point equation in the local potential approximation, a Riccati equation, and the Thomas--Fermi equation. We consider two models where the approach does not apply in order to show the limitations of our Pad\'{e}--Hankel approach.
We investigate the structure of Kubo - Martin - Schwinger (KMS) states on some extension of the universal enveloping algebra of SL(2,C}. We find that there exists a one-to-one correspondence between the set of all covariant KMS states on this algebra and the set of all probability measures d\mu on the real half-line, which decrease faster than any inverse polynomial. This problem is connected to the problem of KMS states on square of white noise algebra.
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