Melnikov, Yuri B. Spectral analysis for a class of integral-difference operators: known facts, new results, and open problems. (English) Zbl 1084.47040 Discrete Dyn. Nat. Soc. 2004, No. 1, 221-249 (2004). The author presents a review of the existing literature on the spectral analysis of integral-difference operators of the form \[ {\mathcal K}_\phi:u(x)\mapsto \int_\infty^\infty \frac{u(x)\phi (s)-u(s)\phi (x)}{| x-s| }ds \] acting on \(L_2({\mathbb R},dx)\), where \(\phi(x)\) is a probability function on \(\mathbb R\). These operators appear in some non-equilibriun statistical physics models as collision operators. Reviewer: Chandra Shekhar Sharma (London) Cited in 6 Documents MSC: 47G20 Integro-differential operators 47N55 Applications of operator theory in statistical physics (MSC2000) 47A10 Spectrum, resolvent 82C99 Time-dependent statistical mechanics (dynamic and nonequilibrium) Keywords:spectral analysis; integral-difference operators; nonequilibrium statistical mechanics PDFBibTeX XMLCite \textit{Y. B. Melnikov}, Discrete Dyn. Nat. Soc. 2004, No. 1, 221--249 (2004; Zbl 1084.47040) Full Text: DOI EuDML