Reparametrizations of Non Trace-normed Hamiltonians

Abstract

We consider a Hamiltonian system of the form y (x) = JH(x)y(x), with a locally integrable and nonnegative 2 × 2-matrix valued Hamiltonian H(x). In the literature dealing with the operator theory of such equations, it is often required in addition that the Hamiltonian H is trace-normed, i.e. satisfies tr H(x) ≡ 1. However, in many examples this property does not hold. The general idea is that one can reduce to the trace-normed case by applying a suitable change of scale (reparametrization). In this paper we justify this idea and work out the notion of reparametrization in detail.

Inverse uniqueness results for one-dimensional weighted Dirac operators

“…In particular, we establish a Borg-Marchenko [5,37] result for this case extending the results from [9]. We also extend some of the results for canonical systems from [48].…”

“…We decided to restrict our considerations to the case of positive definite R since, on the one hand, our main motivation is the Dirac equation (and in this case R = I on (a, b)). On the other hand, a rigorous definition of the operator (linear relation) associated with the spectral problem (2.3) in this case is lengthy (cf., e.g., [22,32,48]), however, the proofs of our main results remain the same.…”

Spectral asymptotics for canonical systems

“…To complete this subsection, let us consider the spectral problem (2.1) also without the trace normalization assumption (2.10). The next result is an analog of Theorem 2.2 (see, e.g., [17,60]) in this case.…”

A Lagrangian View on Complete Integrability of the Two-Component Camassa–Holm System

“…Except for the potential term (that is, the first term on the right-hand side), the equivalent first order system (3.5) has the form of a canonical system; we only mention a small selection of references [1,17,31,35,37,39,41,42]. If the measure υ is absolutely continuous, then it is well known (see, for example, [42,Section 4]) that the system (3.5) can be transformed (by a reparametrization) into an equivalent system with a trace normed weight matrix (that is, the matrix multiplying the spectral parameter on the right-hand side). This is furthermore true in the general case upon slightly modifying the transformation; see [20, Proof of Theorem 6.1].…”