The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by canonical systems, is constant on ℂ. This provides an alternative proof of the De Branges theorem that the canonical systems with tr H1 imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a canonical system.
In this paper, we extend the Remling's Theorem on canonical systems that the ω limit points of the Hamiltonian under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure of a canonical system.
The aim of this paper is to develop the conditions for a symmetric
relation in a Hilbert space ℋ to have self-adjoint extensions in terms of defect
indices and discuss some spectral theory of such linear relation.
The main purpose of this paper is to extend some theory of Schr¨odinger operators from one dimension to higher dimension. In particular, we will give systematic operator theoretic analysis for the Schr¨odinger equations in multidimensional space. To this end, we will provide the detail proves of some basic results that are necessary for further studies in these areas. In addition, we will introduce Titchmarsh- Weyl m− function of these equations and express m− function in term of the resolvent operators.
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