M. Langer and H. Woracek,
Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no
finite generalized poles of non-positive type,
Oper. Matrices 7 (2013), 477–555
Abstract:
The two-dimensional Hamiltonian system
where the Hamiltonian H takes non-negative 2 × 2-matrices as values, and
| ⎛ | 0 | –1 | ⎞ | ||
| J := | ⎜ | ⎟ | , | ||
| ⎝ | 1 | 0 | ⎠ |
has attracted a lot of interest over the past decades. Special emphasis has been put on operator models and direct and inverse spectral theorems. Weyl theory plays a prominent role in the spectral theory of the equation, relating the class of all equations (∗) to the class N0 of all Nevanlinna functions via the construction of Titchmarsh–Weyl coefficients.
In connection with the study of singular potentials, an indefinite (Pontryagin space) analogue of equation (∗) was proposed, where the ‘general Hamiltonian’ is allowed to have a finite number of inner singularities. Direct and inverse spectral theorems, relating the class of all general Hamiltonians to the class N<∞ of all generalized Nevanlinna functions, were established.
In the present paper, we investigate the spectral theory of general Hamiltonians having a particular form, namely, such which have only one singularity and the interval to the left of this singularity is a so-called indivisible interval. Our results can comprehensively be formulated as follows.
- We prove direct and inverse spectral theorems for this class, i.e. we establish an intrinsic characterization of the totality of all Titchmarsh–Weyl coefficients corresponding to general Hamiltonians of the considered form.
- We determine the asymptotic growth of the fundamental solution when approaching the singularity.
- We show that each solution of the equation has ‘polynomially regularized’ boundary values at the singularity.