Spectral theory of the Klein-Gordon equation in Pontryagin spaces
The authors consider the abstract Klein-Gordon equation \[ ((d/dt- iV)^2+ H_0)u= 0, \] where \(H_0\) is a strictly positive self-adjoint operator in a Hilbert space \(H\) with scalar product \((\cdot,\cdot)\) and \(V\) is a symmetric operator in \(H\) such that \(D(H^{1/2}_0)\subset S(V)\), \(S:= VH^{-1/2}_0= S_0+ S_1\) with \(\| S_0\|< 1\) and \(S_1\) compact and \(1\in \rho(S^* S)\). This equation can be transformed into a first-order differential equation, which leads to the operator \[ A:= \begin{pmatrix} 0 & I\\ H & 2V\end{pmatrix},\;H:= H^{1/2}_0(I- S^*S) H^{1/2}_0,\;D(A):= D(H)\oplus D(H^{1/2}_0). \] \(A\) is a self-adjoint operator in the space \(K:= D(H^{1/2}_0)\oplus H\) equipped with the indefinite inner product \(\langle X,X'\rangle:= ((I- S^* S)H^{1/2}_0 x, H^{1/2}_0 x')+ (y,y')\), \(X= (xy)^t\), \(X'= (x'y')^t\in K\). The results of the paper are the following: (1) \(K\) is a Pontryagin space with finite negative integer \(\kappa\), where \(\kappa\) is the number of negative eigenvalues of the operator \(I- S^*S\). (2) \(A\) has a spectral function with at most finitely many critical points. (3) The non-real spectrum of \(A\) is symmetric with respect to the real axis and consists of at most \(\kappa\) pairs of eigenvalues \(\lambda,\overline\lambda\) of finite type. (4) \(\sigma_{\text{ess}}(A)\) is real and \(\sigma_{\text{ess}}\cap(-\alpha, \alpha)= \phi\), where \(\alpha:= (1-\| S_0\|)m\), \(H_0\geq m^2\). (5) \(A\) generates a strongly continuous group of unitary operators in \(K\), which is uniformly bounded. The conditions on the potential \(V\) 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 \(n\leq p<\infty\).
scientific article; zbMATH DE number 5125447
scientific article