Key words Closed extension, M -function, abstract boundary spaces, boundary triplets, elliptic PDEs, pseudodifferential boundary operators, essential spectrum MSC (2000) 35J25, 35J30, 35J55, 35P05, 47A10, 47A11 Dedicated to the memory of Leonid R. VolevichIn this paper, we combine results on extensions of operators with recent results on the relation between the M -function and the spectrum, to examine the spectral behaviour of boundary value problems. M -functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that cannot be detected by the M -function.

We prove that the Dirichelet-to-Neumann map for a Schrödinger operator on a finite simply connected tree determines uniquely the potential on that tree.

The main objective of this paper is to extend the pioneering work of Sims in [9] on secondorder linear differential equations with a complex coefficient, in which he obtains an analogue of the Titchmarsh-Weyl theory and classification. The generalisation considered exposes interesting features not visible in the special case in [9]. An m-function is constructed (which is either unique or a point on a "limit-circle") and the relationship between its properties and the spectrum of underlying maccretive differential operators analysed. The paper is a contribution to the study of non-self-adjoint operators; in general the spectral theory of such operators is rather fragmentary, and further study is being driven by important physical applications, to hydrodynamics, electro-magnetic theory and nuclear physics, for instance.

The paper extends to complex Hamiltonian systems previous work of the authors on the Sims extension of the Titchmarsh–Weyl theory for Sturm–Liouville equations with complex potentials, and analyses the spectral properties of associated non‐self‐adjoint operators. 2000 Mathematics Subject Classification 34B20, 34Lxx.

The problem of integrating the Camassa-Holm equation leads to the scattering and inverse scattering problem for the Sturm-Liouville equation −u + 1 4 u = λwu, where w is a weight function which may change sign but where the left-hand side gives rise to a positive quadratic form so that one is led to a left-definite spectral problem. In this paper the spectral theory and a generalized Fourier transform associated with the equation −u + 1 4 u = λwu posed on a half-line are investigated. An inverse spectral theorem and an inverse scattering theorem are established. A crucial ingredient of the proofs of these results is a theorem of Paley-Wiener type which is shown to hold true. Additionally, the accumulation properties of eigenvalues are investigated.

A new technique is presented which gives conditions under which perturbations of certain base potentials are uniquely determined from the location of eigenvalues and resonances in the context of a Schrödinger operator on a half line. The method extends to complex-valued potentials and certain potentials whose first moment is not integrable.

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For the p-Laplacian ∆p v = div(|∇v| p−2 ∇v), p > 1, the eigenvalue problem −∆p v +q(|x|)|v| p−2 v = λ|v| p−2 v in R n is considered under the assumption of radial symmetry. For a first class of potentials q(r) → ∞ as r → ∞ at a sufficiently fast rate, the existence of a sequence of eigenvalues λ k → ∞ if k → ∞ is shown with eigenfunctions belonging to L p (R n ). In the case p = 2, this corresponds to Weyl's limit point theory. For a second class of power-like potentials q(r) → −∞ as r → ∞ at a sufficiently fast rate, it is shown that, under an additional boundary condition at r = ∞, which generalizes the Lagrange bracket, there exists a doubly infinite sequence of eigenvalues λ k with λ k → ± ∞ if k → ± ∞. In this case, every solution of the initial value problem belongs to L p (R n ). For p = 2, this situation corresponds to Weyl's limit circle theory.