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In this paper the spectral properties of the abstract Klein–Gordon equation are studied. The main tool is an indefinite inner product known as the charge inner product. Under certain assumptions on the potential V, two operators are associated with the Klein–Gordon equation and studied in Krein spaces generated by the charge inner product. It is shown that the operators are self-adjoint and definitizable in these Krein spaces. As a consequence, they possess spectral functions with singularities, their essential spectra are real with a gap around 0 and their non-real spectra consist of finitely many eigenvalues of finite algebraic multiplicity which are symmetric to the real axis. One of these operators generates a strongly continuous group of unitary operators in the Krein space; the other one gives rise to two bounded semi-groups. Finally, the results are applied to the Klein–Gordon equation in ℝn.

Let A be a self-adjoint operator in a Krein space $(\mathcal{K},[ \cdot , \cdot ]).$ Under certain natural assumptions, it is shown precisely which real eigenvalues of A can be given a max-inf characterization generalizing the usual one in Hilbert space. This result unifies several approaches in the recent literature.

LetH=(A, B) be a pair of HermitianN×N matrices. A complex number λ is an eigenvalue ofH ifdet(A−λB)=0 (we include λ=∞ ifdetB=0). For nonsingularH (i.e., for which some λ is not an eigenvalue), we show precisely which eigenvalues can be characterized as σk+=sup{inf{χ*Aχ:χ*Bχ=1,χ∈S},S∈Sk},Sk being the set of subspaces of CN of codimensionk−1.

We consider the second order initial value problem in a Hilbert space, which is a singular perturbation of a first order initial value problem. The difference of the solution and its singular limit is estimated in terms of the small parameter ∈ The coefficients are commuting self-adjoint operators and the estimates hold also for the semilinear problem.

LetA be a selfadjoint definitizable operator in a Krein space. It is shown that there exists a finite rank nonnegative perturbation ofA of arbitrarily small norm such that all the singular critical points ofA of finite index disappear.

A sufficient condition for the stability of the range of a positive operator in a Hilbert space is given. As a consequence, we get a class of additive perturbations which preserve regularity of the critical point 0 of a positive operator in a Krein space.

The norm of the inverse operator of ϵA + B - λI between the Besov spaces B, ∞ (Ω) and Bt∞,∞(Ω) is estimated, where A and B are uniformly elliptic operators with smooth coefficients and Dirichlet boundary conditions, A is of order 2m, B of order 2m, m > m'. The estimate holds for negative t. The Besov space B, ∞ (Ω) reduces to the space of Holder continuous functions Cs (Ω) if s < 0 is non – integer. In particular, it is shown that Aϵ generates an analytic semigroup in B, ∞ (Ω), s ϵ (-1,0), if Ω = ℝn or ℝ and A, B are constant coefficient operators without lower order terms.

Let A be a self-adjoint operator on a finite dimentional inner product space K which is nondegenerate (i.e., K ∩ K ⊥ = 0). For a fixed Jordan basis B of A, we examine sets of Jordan chains in B which span nondegenerate subspaces. An example is the set of all chains in B of any given set of lengths.

A variational characterization, involving a max-inf of the Rayleigh quotient, is established for certain eigenvalues of a wide class of definitizable selfadjoint operators Q in a Krein space. The operator Q may have continuous spectrum and nonreal and nonsemisimple eigenvalues; in particular it may have embedded eigenvalues. Various applications are given to selfadjoint linear and quadratic eigenvalue problems with weak definiteness assumptions.

This note contains some supplements to our earlier notes [LN II], [LN III], where the Newton diagram was used in order to obtain in a straightforward way information about the perturbed eigenvalues of an analytic and analytically perturbed matrix function.

As in [N], [LN] the Newton diagram is used in order to get information about the first terms of the Puiseux expansions of the eigenvalues λ(e) of the perturbed matrix pencilT(λ, e)=A(λ)+B(λ, e) in the neighbourhood of an unperturbed eigenvalue λ(∈) ofA(λ). In fact sufficient conditions are given which assure that the orders of these first terms correspond to the partial multiplicities of the eigenvalue λ0 ofA(λ).

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 < ∞.

Let A and B be selfadjoint operators ona Hilbert space H. We define minimal index ν(A,B) = min {#negative eigenvalues of A - λB}, we connect it with various ideas in the literature and we connect it with formulae used in some recent variational principles.

We consider a nonnegative selfadjoint operator A in a Krein space such that p(A) ≠ θ and ∞ ∉ c s (A) (i.e. ∞ is not a singular critical point of A). Then we show that these properties remain true for a certain perturbation of the operator, acting in a slightly perturbed Krein space. This result is applied to elliptic differential operators with indefinite weights and to certain difference operators.

We characterize a class of indefinite partial differential operators which are similar to selfadjoint operators in the Hilbert space L 2(ℝ n ).

For two different examples of positive definitizable operators in Krein spaces we prove regularity of the critical points 0 and ∞. The first example is the Sturm-Liouville operator in L 2(ℝ,ρ) with ρ(x) = 007C;x007C; s sgn x. The second example is a difference operator in a Krein space of complex sequences.

It is shown that the operators associated with the perturbed wave equation in ℝn and with elliptic operators with an indefinite weight function and mildly varying coefficients on ℝn are similar to a self-adjoint operator in a Hilbert space. These operators have the whole ℝn as the spectrum. It is shown that they are positive operators in corresponding Krein spaces, and the whole problem is reduced to showing that 0 is not a singular critical point.

Consider the weighted eigenvalue problem $$ Lu = {\rm \lambda }\left( {{\rm sgn}\;{\rm x}} \right)u, $$ (1)

Following the work of Krein and Langer [11], the spectral function has become a basic tool in the study of self-adjoint operators in Krein spaces, cf. [2, §4.1], [5, §VIII 6], [12] and the references therein. The resulting spectral decompositions behave as in Hilbert space except near the (at most finite) set of critical points, where the spectral function is, at least initially, undefined. Our purpose is to study these critical points in situations that permit finite dimensional analysis. In particular, we shall give a number of finite dimensional tests for regularity of critical points.

In [5], [7] is was shown that the operator of KLEIN-GORDON type is a spectral operator with critical points. In this paper we estimate where the critical points can be found. We are especially interested in finding sufficient conditions for the absence of singular critical points.

A classical theorem of Cauchy states that the eigenvalues of a principal submatrix A 0 of a Hermitian matrix A interlace the eigenvalues of A . We consider the case of a matrix A which is Hermitian with respect to an indefinite inner product.

Consider the Klein-Gordon equationwhere q, Aj are real valued functions on Rn, m and e positive constants. Equation (1) describes the motion of a relativistic particle of mass m and charge e in an external field described by the electrostatic potential q and the electromagnetic potential A = (Aj); units are chosen so that the speed of light is one.

Let A be a uniformly elliptic second order linear operator on a smooth bounded domain Ω⊂ R n. We study the eigenvalue problem Au=λu subject to boundary conditions B0u=λB1u on ∂Ω, where Bj are linear boundary operators. The problem is recast in the form A u=λu in a Hilbert or Krein space, and results are given on the location and type of the spectrum, full- and half-range completeness, and regularity of critical points.

We establish a minimax characterization for extreme real eigenvalues of a general hermitian pencil λA − B. The matrix A is allowed to be singular, so infinity may be an eigenvalue. It is also proved that the extremum can be taken over real subspaces if A and B are real.

Second order differential equations in Hilbert spaces motivate the investigation of matrix operators with unbounded entries. Sufficient conditions are given for the closure of such an operator to be a generator of a C o -semigroup or a C -group. The abstract result is applied to the Klein-Gordon equation

According to Mathon and Rosa /The CRC Handbook of combinatorial designs, CRC Press, 1996/ there is only one known symmetric design with parameters (69, 17, 4). This known design is given in Beth, Jungnickel, and Lenz /Design theory, B.I. Mannheim, 1985/ ; the Frobenius group F39 of order 39 acts on this design, where Z13 has exactly 4 fixed points and Z3 has exactly 9 fixed points. The purpose of this article is to investigate the converse of this fact with the hope of obtaining in this way at least one more design with these parameters. In fact we obtain exactly one new such design. In this article we have classified all such designs invariant under F39.

A symmetric design with parameters (115,19,3) is the smallest triplane of the quadratic order, the existence of which is in doubt.Investigations on the possibility tha the group A5 operates as an automorphism group on a symmetric design with parameters (115,19,3) have been made with an intention of constructing such a design.The results of this paper are derived under the condition that an automorphism of order three has the minimal allowed number of fixed points.

Symmetric designs with parameters (v,k,l) which have the square order n=k-l are of special interest because they could have very large automorphism groups. The starting point are the two results which have been recently proved.