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Summary: On the Space of Symmetric Operators with
Multiple Ground States
A. Agrachev
Abstract
We study homological structure of the filtration of the space of self-
adjoint operators by the multiplicity of the ground state. We consider
only operators acting in a finite dimensional complex or real Hilbert
space but infinite dimensional generalizations are easily guessed.
1 Introduction
This paper is dedicated to the memory of V. I. Arnold and is somehow
inspired by his works [1], [2] (see also [5]). It opens a planned series of
papers devoted to homological invariants of the families of quadratic forms
and related geometric structures; Theorem 2 below forms a fundament of all
further development.
In this paper we study the filtration of the space of self-adjoint operators
by the multiplicity of the ground state. We restrict ourself to the operators
in a finite dimensional complex or real Hilbert space, but possible infinite
dimensional generalizations are easily guessed.
Let 1(A) 2(A) · · · k(A) · · · be ordered eigenvalues of the
operator A. The operators with the ground state of multiplicity at least k
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