This paper is devoted to the spectral theory (more precisely, tothe variational theory of the spectrum) of guided waves in anelastic half cylinder.  We use variational methods to investigateseveral aspects of propagating waves, including localization (seeFigure 1), existence criteria and the formulas to find them. Weapproach the problem using two complementary methods: Thevariational methods fo...

‎in this paper‎, ‎we study spectral element approximation for a constrained‎ ‎optimal control problem in one dimension‎. ‎the equivalent a posteriori error estimators are derived for‎ ‎the control‎, ‎the state and the adjoint state approximation‎. ‎such estimators can be used to‎ ‎construct adaptive spectral elements for the control problems.

This paper is devoted to the proof of the unique solvability ofthe inverse problems for second-order differential operators withregular singularities. It is shown that the potential functioncan be determined from spectral data, also we prove a uniquenesstheorem in the inverse problem.

We prove, assuming that the Bohr-Sommerfeld rules hold, that the joint spectrum near a focus-focus critical value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to ~-pseudodifferential operators, and to Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.

We continue the study of the A-amplitude associated to a half-line Schrödinger operator, − d dx2 + q in L2((0, b)), b ≤ ∞. A is related to the WeylTitchmarshm-function via m(−κ2) = −κ− ∫ a 0 A(α)e −2ακ dα+O(e−(2a−ε)κ) for all ε > 0. We discuss five issues here. First, we extend the theory to general q in L1((0, a)) for all a, including q’s which are limit circle at infinity. Second, we prove th...