# zbMATH — the first resource for mathematics

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
Full Text: