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Spectral theory of the Klein-Gordon equation in Pontryagin spaces. (English) Zbl 1114.47038

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 \(n\leq p<\infty\).

MSC:

47B50 Linear operators on spaces with an indefinite metric
47B25 Linear symmetric and selfadjoint operators (unbounded)
35P05 General topics in linear spectral theory for PDEs
35Q40 PDEs in connection with quantum mechanics
47A10 Spectrum, resolvent
47F05 General theory of partial differential operators
47N50 Applications of operator theory in the physical sciences
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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