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.


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