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Some Open Problems

At conferences I like to ask people "If a Fairy offers you the key idea for proving a Theorem of your choice, what would you pick?". It is the information you don't usually find in published papers. The following sites collect such material. Below you find my own modest and somewhat random collection. The posting date is given in brackets. Dates earlier than 1997 were originally posted at may former home page in Osnabrück.

So if any of these ring a bell, or you know of partial results LET ME KNOW!. If any problem gets solved, I will also place a link to the solution.

tr exp (A+itB) [Nov. 1996]
A conjecture of Bessis, Moussa, and Villani (J.Math.Phys.16(1975) 2318-2325) says that, for arbitrary hermitian matrices A and B the function t-->tr exp (A+itB) is the Fourier transform of a positive measure. This would entail a number of interesting inequalities not just for quantum partition functions, but also for their dervatives (a badly needed tool). Despite a lot of work, some by prominent mathematical physicists, only some simple cases have been decided. So far all results, including fairly extensive numerical experiments, are in agreement with the conjecture. I have been collecting material on this problem with Mark Fannes (along with our own work).
Uniqueness of q-relations [Nov. 1996]
This refers to the relations I studied with Palle Jørgensen and Lothar Schmitt (items 42 and 47 of the publication list). We proved the basic uniqueness result for q satisfying |q| <\sqrt{2}-1, but this is certainly not optimal. The problem is to find the best interval for this result, which is possibly |q|<1. This problem is described at some length in Ref.47.
A Conjecture by R.S. Phillips [Nov. 1996]
A Krein space is an indefinite scalar product space, which is topologically a Hilbert space. To anyone mathematically brought up in Hilbert space, a Krein space is a terrible place to be. Nevertheless, they come up naturally even in some honest Hilbert space problems, notably the extension theory of symmetric operators. To solve a classification problem related to canonical commutation relations and "arrival time operators" (see paper) I need an answer to the following question. The answer "yes" was conjectured by Phillips in 1960
Consider a one-parameter (Krein-)unitary group in a Krein space. Among the many subspace which are maximal with respect to the property that the indefinite scalar product on them ("maximal positive subspaces"), can we find one which is invariant under the given group? The answer is easy, if the dimension of a maximal positive (or maximal negative) subspace is finite. Otherwise, I don't know.
Last updated: Sept 30, 1999
r.werner@tu-bs.de