A trace formula for canonical differential expressions.
The authors study \(2n \times 2n\) canonical differential expressions of the form \[ - i J \frac{d}{dt}f(t,z)=z\,f(t,z) + V(t)\,f(t,z), \quad t \geq 0, \;z \in {\mathbb C}, \] where \(J\) is the block diagonal matrix with entries \(I_n\) and \(-I_n\) and \(V\) is a block anti-diagonal matrix function with entries \(k\) in the right upper and \(k^\ast\) in the left lower corner, \(k\) being an \(n\times n\)-matrix function the entries of which are complex-valued \(L_1( 0,\infty)\) functions. With such a differential expression, two selfadjoint operators \(H_+\) and \(H_-\) are associated, which are formally given by\(H_\pm=-\text{ i} J \frac{d}{dt} \pm V\). It is proved that for every nonreal \(z\) the operator \((H_+-z)^{-1}-(H_--z)^{-1}\) has rank \(n\) and a formula for its trace is derived in terms of the corresponding Weyl function.
scientific article; zbMATH DE number 1895832
scientific article