Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M‐function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M‐function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M‐function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M‐function corresponds to the Dirichlet to Neumann map.

Abstract. In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M -function see the same singularities as the resolvent of a certain restriction A B of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces S andS such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M -function is analytic. We present three examples -one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line -which together indicate that the abstract results are probably best possible.

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We study the essential spectrum of operator pencils associated with anisotropic Maxwell equations, with permittivity ε, permeability µ and conductivity σ, on finitely connected unbounded domains. The main result is that the essential spectrum of the Maxwell pencil is the union of two sets: namely, the spectrum of the pencil div((ωε+iσ)∇ ⋅ ), and the essential spectrum of the Maxwell pencil with constant coefficients. We expect the analysis to be of more general interest and to open avenues to investigation of other questions concerning Maxwell's and related systems.

We prove some new results which justify the use of interval truncation as a means of regularising a singular fourth order Sturm-Liouville problem near a singular endpoint. Of particular interest are the results in the so called lim-3 case, which has no analogue in second order singular problems.Definition 2.3 An endpoint is said to be of lim-p type if the space of solutions of the differential equation ℓy = iy which are square integrable at that endpoint has dimension p.For fourth order Sturm Liouville problems, the only possibilities are p = 2, p = 3, and p = 4. The lim-2 case is analogous to the limit-point case for second order equations; the lim-4 case is analogous to limit-circle. Lim-3 has no second order analogue. Regular endpoints are of lim-4 type. These endpoint types should not be confused with the deficiency indices of the minimal operator: for example, if both endpoints are of lim-2 type then the deficiency indices are zero.Boundary conditions are imposed by means of the Lagrangian form, which we now define.