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Spectral analysis for a class of integral-difference operators: known facts, new results, and open problems. (English) Zbl 1084.47040

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.

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)
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