Research description

Ex astris, scientia.

1. Szegö limit theorems In quantum mechanics computing transition probabilities in many-fermion systems leads to section determinants of linear operators. Here the term section determinant is to be understood as the determinant of a linear operator evaluated with respect to a finite dimensional subspace instead of the entire space on which the operator acts. Depending on what is the concrete physical question different kinds of operators occur. Here two types are of special importance and interest. These are semi-groups and spectral projections.
   
   1. Section determinants of semi-groups appear when posing the question with what probability a many-fermion system returns into its original state. In [3], which is an improved version of parts of [2], an integral formula that expresses the section determinants through the solution to a linear integral equation is proved. The unique solvability of this equation turns out to be equivalent to the non-vanishing of the determinant. In order to be useful for practical purposes solvability criteria are needed that do not involve the determinant. Such a criterion can be derived for the special case of unitary operators.
      Two problems arise. On the one hand one can look for other sufficient criteria that ensure the non-vanishing of the determinant. An answer to this question is indicated in the criterion that has been derived already. On the other hand this also motivates the search for necessary criteria, i.e. for classes of operators that do not generally admit a solution to the integral equation.
      The integral formula can also be used for quantitative investigations. Here, especially, the long-time asymptotics is of interest. One may think of using the integral equation for this purpose.
      Alternatively, section determinants can be described by a formula that relates them to the solutions to a non-linear differential equation, the so-called (operator-valued) Riccati equation. This was done in [5] where an a-priori estimate for solutions to a special initial value problem for this differential equation is obtained. This ensures the existence of a globally defined solution.
      Though the same problems arise as before here the emphasis does not lie on deriving other criteria but on strenghtening the conclusions that can be drawn from the present criterion. Therefore, it seems to be necessary to refine both the analytical and algebraic tools. The latter ones seem to be in some analogy to the celebrated Campbell-Baker-Hausdorff formula. These investigations are aimed at Szegö type theorems for the section determinants of unitary operators and at providing a functional analytic proof of the classical theorem.
      
      
   2. When a many-fermion system is exposed to a sudden pertubation the question about the transition probability is mathematically reflected by computing the section determinant of spectral projections of self-adjoint operators. In [2] and [6] a variant of the above integral formula is proved, which can also be interpreted as a variant of the classical adiabatic theorem.
      As before, the integral formula connects the section determinants with the solution to a linear integral equation, namely a Wiener-Hopf equation. Solvability of the integral equation and non-vanishing of the determinant are again related but not equivalent without further assumptions. Accordingly, only a uniqueness criterion that does not use the determinant can be derived.
      In the proof of the integral formula two gap conditions were used, which are related to the numerical range and the spectrum of the self-adjoint operator involved. For physical reasons it is important to know how these conditions govern the quantitative behaviour. This special question also provides a link to inertia theory of linear operators and to higher order spectra of self-adjoint operators.
      As above, the hope is to end up with a Szegö type theorem for spectral projections, which shows the exact many-particle asymptotics and which would find applications to the so called X-ray-edge effect and the Anderson orthogonality catastrophe.
      In some sense the integral formula for spectral projections can be considered the limiting case of the integral formula for semi-groups as the physical time tends to infinity. Therefore, studying section determinants of semi-groups with emphasis on long time asymptotics might be useful for spectral projections, too.
      
      
   3. Since section determinants naturally appear in the context of fermionic systems it seems quite obvious to utilize the abstract Fock space setting in which many-particle problems are most elegantly formulated. There are indeed some hints that the celebrated Borodin-Okounkov formula, which relates Toeplitz determinants with Hankel determinants, might be regarded as a bosonization relation. A rigorous proof would not only be desirable from a mathematical point of view but could also forebode possible generalizations.
   
   
2. Hankel Matrices When studying Szegö theorems for Hankel matrices one is also concerned with their spectral theory. The most famous such matrix is the Hilbert matrix which has been the object of intense study. It can be written as integral operator and then explicetly diagonalized in terms of special functions. However, a direct approach using the matrix representation would be desirable. An application is asymptocially precise bounds on the spectral radius of the finite Hilbert matrix which would improve Hilbert's inequality and yield a direct proof and possibly an improvement of the de Bruijn-Wilf formula.
   
   
3. Self-adjointness and non-self-adjointness of Jacobi matrices Computing the deficiency indices of operators, that are related to the harmonic oscillator, leads to Jacobi or Jacobi-like matrices. The classical criteria like, e.g. Carleman and Berezansky do not suffice to decide whether or not these matrices are self-adjoint. A perturbation theoretic approach does not work either though the related estimates (Hardy type inequalities) may be interesting in its own right. Anyway, new and refined criteria are needed.