A Krein space approach to symmetric ordinary differential operators with an indefinite weight function

A Krein Space Approach to Symmetric Ordinary Differential Operators with an Indefinite Weight Function

l(f)=(-l)“(Pof’“‘)‘“‘+(-l)“-‘(p,f’”-”)’”~”+ ... +p,f=A.rf (0.1) on a finite or infinite interval (a, b) with real, locally summable coefficients l/PO,Pl, ..-3 Pnv r under the assumptions that p0 >O and that the weight function r changes its sign on (a, b). If r is positive, problem (0.1) can be studied in the context of Hermitian and self-adjoint operators in the Hilbert space L*(r) with the in...

Definitizable Extensions of Positive Symmetric Operators in a Krein Space

The Friedrichs extension and the Krein extension of a positive operator in a Krein space are characterized in terms of their spectral functions in a Krein space.

A Krein Space Approach to Elliptic Eigenvalue Problems with Indefinite Weights

where L is a symmetric elliptic operator and r is a locally integrable function on Rn. If r is of constant sign then this problem leads to a selfadjoint problem in the Hilbert space L2(|r|). In this note we are interested in the case when r takes both positive and negative values on sets of positive measure. Then the spectrum of the problem (1) is not necessarily real any more. Moreover it is n...

Algebra with Indefinite Involution and Its Representation in Krein Space

It is often inevitable to introduce an indefinite-metric space in quantum field theory, for example, which is explained for the sake of the manifestly covariant quantization of the electromagnetic field. We show two more evident mathematical reasons why such indefinite metric appears. The first idea is the replacement of involution on an algebra. For an algebra A with an involution † such that ...

Spectral properties of non - symmetric systems of ordinary differential operators