Abstract. For Sturm-Liouville operators on the half line, we show that the property of having singular, singular continuous, or pure point spectrum for a set of boundary conditions of positive measure depends only on the behavior of the potential at infinity. We also prove that existence of recurrent spectrum implies that of singular spectrum and that "almost sure" existence of L 2 -solutions implies pure point spectrum for almost every boundary condition. The same results hold for Jacobi matrices on the discrete half line. §1 Introduction For Sturm-Liouville operators generated by − d 2 dx 2 +q on the half line [0, ∞), we study the dependence of spectral types on the boundary condition at 0 and on compactly supported perturbations of the potential. In Weidmann [17], it was conjectured that the existence of singular spectrum depends only on the behavior of the potential close to infinity. This, strictly speaking, is not true (see [4,5]). However, we now prove that existence of singular, singular continuous, or pure point spectrum for a set of boundary conditions of positive measure does not depend on the local behavior of the potential ( §5).Our proof of this result is prepared in §2-3 and relies mainly on:(i) The identification of Aronszajn [1] and Donoghue [7] of the various parts of the spectrum under variation of boundary condition or rank one perturbation. (ii) A result on the average of the spectral measure with respect to boundary condition that goes back to Javrjan [9]; we use it in a form rediscovered by Kotani [12]. (iii) The invariance of the absolutely continuous spectrum under local perturbations. (iv) Invariance of the set of energies with solutions L 2 at infinity under local perturbations.

We study how the spectrum of a Jacobi operator changes when this operator is modified by a certain finite rank perturbation. The operator corresponds to an infinite mass‐spring system and the perturbation is obtained by modifying one interior mass and one spring of this system. In particular, there are detailed results of what happens in the spectral gaps and which eigenvalues do not move under the modifications considered. These results were obtained by a new tecnique of comparative spectral analysis and they generalize and include previous results for finite and infinite Jacobi matrices.

We study invariance for eigenvalues of selfadjoint Sturm-Liouville operators with local point interactions. Such linear transformations are formally defined byor similar expressions with δ ′ instead of δ. In a probabilistic setting, we show that a point is either an eigenvalue for all ω or only for a set of ω's of measure zero. Using classical oscillation theory it is possible to decide whether the second situation happens. The operators do not need to be measurable or ergodic. This generalizes the well known fact that for ergodic operators a point is eigenvalue with probability zero.

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