On 2/25/14, Veit Elser <ve10@cornell.edu> wrote:
Besides having used the term in a publication, I'm a native German speaker and have had lunch with Bethe.
When used properly it has a precise meaning. The context is always an equation whose solution is being sought. It can be a differential equation, a functional equation, or really any kind of equation for which there is not an established body of theory for finding the most general solution. A solution "Ansatz" is simply a proposed form of the solution, with no guarantees that the form is correct. Of course you don't see papers where an Ansatz is proposed only to have it fail the test of being a solution. Bethe's Ansatz is a proposal for the form of the eigenvectors of certain "transfer matrices" that arise in statistical mechanics. It acquired quite a reputation when the same form, or Ansatz, proved to be successful in a large class of models in two dimensions.
-Veit
Light dawns --- Lieb assumes a certain form for the eigenvector; if it then proves possible to solve the resulting linear equations for an eigenvalue, then the assumption becomes retrospectively justified. Of course, all this can only happen in the limit as m -> oo , since for finite m > 3 the relevant eigenvalues are algebraic. Also we need to know that we found the maximum --- this seems instead to be what is being addressed at the start of section IV . So the "ansatz" has no connection with justifying the assumption that the max eigenvalue actually contributes to the counting function. This disillusions me even more ... I'm now uncertain whether the issue has even occurred to him! Fred Lunnon