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Math. Phys. 1007/s00220-010-1177-6 Communications in Mathematical Physics Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials A. Martínez-Finkelshtein1,2 , E. A. Rakhmanov3 1 Department of Statistics and Applied Mathematics, University of Almería, 04120 Almeria, Spain. es 2 Instituto Carlos I de Física Teórica y Computacional, Granada University, 18071 Granada, Spain 3 Department of Mathematics, University of South Florida, Tampa, FL 33620, USA. edu Received: 6 April 2009 / Accepted: 25 July 2010 Published online: 8 January 2011 – © Springer-Verlag 2011 Abstract: We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials – polynomial solutions of second order differential equations with complex polynomial coefficients.

21, λ−1 ∂ y Rλ behaves like the sum of Rλ and a pole term of form λ−1 e−θ|x−y| . 10) up to a constant (independent of N ) term of the form Ce−θ|x−y| , where the space-time derivatives of C are uniformly bounded. This constant term satisfies the same bound as term I. 64)(ii). By dominated convergence, H1 − lim N →∞ R E(ε, N T ) f (y) dy exists and is equal to a sum of terms of the form C(ε, T )(U¯ ε ) R f (y) dy. 64), R (S + R)(ε, N T ) f (y) dy H1 ≤ C(N T )− f rac14 f L1 . 12) This proves convergence in H 1 of the sequence II N .

We now carry out a nonstandard Lyapunov–Schmidt reduction following the “inverse temporal dynamics” framework of [TZ2], tailored for the situation that Id − S(ε, T ) is not uniformly invertible, or, equivalently, the spectrum Transition to Longitudinal Instability of Detonation Waves 41 of (1 − )L(ε) is not bounded away from { jπ/T } j∈Z . In the present situation, (1 − )L(ε) has both a 1-dimensional kernel (a consequence of (H4), see Sect. 5) and essential spectra accumulating at λ = 0, and no other purely imaginary spectra, so that Id − S(ε, T ) inherits the same properties; see [TZ2] for further discussion.

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