 
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 selfadjoint 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
