|
VIDEO LIBRARY |
International youth conference "Geometry & Control"
|
|||
|
Discrete Dynamics of the Tyurin Parameters and Commuting Difference Operators Gulnara S. Mauleshova, Andrey E. Mironov Sobolev Institute of Mathematics, Novosibirsk, Russia |
|||
Abstract: We study commuting difference operators of rank two. In the case of hyperelliptic spectral curves an equation which is equivalent to the Krichever – Novikov equations on Tyurin parameters is obtained. With the help of this equation examples of operators corresponding to hyperelliptic spectral curves of arbitrary genus are constructed. Among these examples there are operators with polynomial and trigonometric coefficients. If two difference operators $$ L_4=\sum^{2}_{i=-2} u_i(n)T^i, \qquad L_{4g+2}=\sum^{2g+1}_{i=-(2g+1)}v_i(n)T^i,\qquad u_2=v_{2g+1}=1 $$ commute, where $$ \Gamma=\{(z,w)\in {\mathbb C}^2| F(z,w)=0\}. $$ The common eigenvalues are parametrized by the spectral curve $$ L_4\psi=z\psi, \quad L_{4g+2}\psi=w\psi, (z,w)\in \Gamma. $$ The rank of the pair $$ l={\rm dim}\{\psi:L_4\psi=z\psi, \ \ L_{4g+2}\psi=w\psi,\ \ (z,w)\in \Gamma.\} $$ The curve $$\sigma:\Gamma\rightarrow\Gamma,{\ }{\ }{\ }\sigma(z,w)=\sigma(z,-w).$$ The common eigenfunctions $$ \psi_{n+1}(P)=\chi_1(n,P)\psi_{n-1}(P)+\chi_2(n,P)\psi_n(P), $$ The functions The following theorems are proved. Theorem 1. If $$ \chi_1(n,P)=\chi_1(n,\sigma(P)),\qquad \chi_2(n,P)=-\chi_2(n,\sigma(P)), $$ then $$ L_4=(T+V_nT^{-1})^2+W_n, $$ herewith $$ \chi_1=-V_n\frac{Q_{n+1}}{Q_{n}},\qquad \chi_2=\frac{w}{Q_n}, $$ where $$ Q_n(z)=z^g+\alpha_{g-1}(n)z^{g-1}+\ldots+\alpha_0(n). $$ Functions $$ F_g(z)=Q_{n-1}Q_{n+1}V_n+Q_{n}(Q_{n+2}V_{n+1}+Q_{n+1}(z-V_n-V_{n+1}-W_n)). $$ Theorem 2. The operator $$ L_4=(T+(r_3n^3+r_2n^2+r_1n+r_0)T^{-1})^2+g(g+1)r_3n $$ commutes with a difference operator Theorem 3. The operator $$ L_4=(T+(r_1a^n+r_0)T^{-1})^2+(a^{2g+1}-a^{g+1}-a^g+1)r_1a^{n-g} $$ commutes with a difference operator Theorem 4. The operator $$ L_4=(T+(r_1\cos(n)+r_0)T^{-1})^2-4r_1\sin(\frac{g}{2})\sin(\frac{g+1}{2})\cos(n+\frac{1}{2}) $$ commutes with a difference operator Language: English References
|