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The spectral theory of a functional-difference operator in conformal field theory. (English. Russian original) Zbl 1327.39013
Izv. Math. 79, No. 2, 388-410 (2015); translation from Izv. Ross. Akad. Nauk, Ser. Mat 79, No. 2, 181-204 (2015).
The authors consider the operator \[ (H\psi)(x)=\psi\left(x+2\omega'\right)+\psi\left(x-2\omega'\right)+e^{\frac{\pi ix}{\omega}}\psi(x), \] where \(\omega\) and \(\omega'\) are purely imaginary numbers, satisfying \(\operatorname{Im}(\omega)>0\) and \(\operatorname{Im}(\omega')>0\). The function \(x\mapsto\psi(x)\) is assumed to be analytic in the strip \(|\operatorname{Im}(z)|\leq2|\omega'|\). The authors then conduct an analytic study of the functional-difference operator \(H\). As part of this study, the authors study the scattering theory for \(H\) as well as provide an eigenfunction expansion theorem.

MSC:
39A70 Difference operators
47B39 Linear difference operators
47A10 Spectrum, resolvent
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
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