Spectral Theory of the Klein–Gordon Equation in Pontryagin Spaces

Abstract

In this paper we investigate an abstract Klein–Gordon equation by means of indefinite inner product methods. We show that, under certain assumptions on the potential which are more general than in previous works, the corresponding linear operator A is self-adjoint in the Pontryagin space $$\mathcal{K}$$ induced by the so-called energy inner product. The operator A possesses a spectral function with critical points, the essential spectrum of A is real with a gap around 0, and the non-real spectrum consists of at most finitely many pairs of complex conjugate eigenvalues of finite algebraic multiplicity; the number of these pairs is related to the ‘size’ of the potential. Moreover, A generates a group of bounded unitary operators in the Pontryagin space $$\mathcal{K}$$ . Finally, the conditions on the potential required in the paper are illustrated for the Klein–Gordon equation in $$\mathbb{R}^n$$ ; they include potentials consisting of a Coulomb part and an L p -part with n ≤ p < ∞.