arXiv:1807.09252v1 [math.AP] 24 Jul 2018 On the commutation of finite convolution and differential operators Yury Grabovsky, Narek Hovsepyan Abstract We study those commutation relations between finite convolution integral operator K and differential operators, that have implications for spectral properties of K. This includes classical commutation relation KL = LK, as well as new commutation rela- tions, such as KL 1 = L 2 K. We obtain a complete characterization of finite convolution operators admitting the generalized commutation relations. Contents 1 Introduction 2 2 Main Results 5 2.1 Commutation ................................... 5 2.2 Sesqui-commutation ............................... 8 3 Commutation 10 4 Sesqui-commutation 12 4.1 Reduction of the general case .......................... 15 4.2 L 1 = L 2 ...................................... 17 4.2.1 Equation for k(z), boundary conditions ................. 17 4.2.2 Single mode and multiplicities ...................... 18 4.2.3 Multiple modes .............................. 22 4.2.4 Item 1, γ =0 ............................... 28 4.2.5 Item 3 ................................... 28 4.3 L 2 = −L 1 ..................................... 30 5 Appendix 31 5.1 Forms of a, b and c ............................... 31 5.2 Reduction ..................................... 32 5.3 Finding k ..................................... 36 1