Perturbations of elliptic operators on high codimension subsets and the extension theory on an indefinite metric space

The spectral aspect of the problem of perturbations supported on thin sets of codimension θ≥2m in ℝⁿ is considered for elliptic operators of order m. The problem of realization of such perturbations is formulated as a problem of self-adjoint extension of a linear symmetric relation in a space with indefinite metric. It is shown how to construct such a relation for a given elliptic operator and a family of distributions. Its functional model is obtained in terms of Q-functions. Self-adjoint extensions and their resolvents are described. The theory developed is applied to quantum models of point interactions in high dimensions and high moments. Bibliography: 35 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1900, pp. 246–292.     

Spectral inclusion properties of the numerical range in a space with an indefinite metric

In this paper the numerical range of operators (possibly unbounded) in an indefinite inner product space is studied. In particular, we show that the spectrums of bounded positive operators (or the spectrum of unbounded uniformly I-positive operators) are contained in the closure of the I-numerical range.     

Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space

The problem of realization of nontrivial perturbations supported on thin sets of codimension in Rⁿ for elliptic operators of order m, when 2m, is formulated as one of construction of the self-adjoint extensions of some symmetric linear relation in an indefinite metric space. The self-adjoint extensions and their resolvents are described. It is found that the same extensions can be obtained as a result of extensions of some symmetric operator in L₂ (Rⁿ) with it going out to a larger indefinite metric space. But such an operator is chosen already by the nonlocal boundary conditions. Applications to quantum models of point interactions are discussed.In Memory of Mikhail Constantinovich PolivanovV. A. Steklov Mathematical Institute, Russian Academy of Sciences. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 466–472, September, 1992.     

The eikonal equation of an indefinite metric

The concept of a semi-Riemannian map is introduced and it is shown that such maps are solutions of the eikonal equation. Also the existence of solutions to the eikonal equation are discussed and their relation to the Laplace-Beltrami equation is investigated.Supported by the project TBAG-CG2, Tübitak, Turkey.     

Unitarity condition in covariant quantum field theory with indefinite metric

V. A. Steklov Mathematics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 3, pp. 347–358, June, 1989.     

On M. G. Krein's papers in the theory of spaces with an indefinite metric

This is a survey of the papers by M. G. Krein (and his disciples) devoted to the theory of operators in spaces with an indefinite metric and its applications.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 5–17, January–February, 1994.