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SOME APPLICATIONS OF WEYL CALCULUS TO BURCHNALL-CHAUNDY THEORY, III
Keywords:
Weyl calculus, Kohn-Nirenberg calculus, differential operatorsDOI:
https://doi.org/10.17654/0972087125002Abstract
We show how Weyl calculus can be applied to the following problem. Suppose that $A, B$ are differential operators with polynomial coefficients and they commute. Then there is a polynomial $f(\lambda, \mu)$ such that $f(A, B)=0$. We use Weyl calculus, but Kohn-Nirenberg calculus also can be used. In this paper, we give the proof of existence of $f(\lambda, \mu)$ in general case, prove that the ideal $J$ of all such $f(\lambda, \mu)$ is the principal ideal, and give an algorithm for finding generator of $J$ in some particular cases.
Received: August 4, 2024
Accepted: September 28, 2024
References
A. Voros, An algebra of pseudodifferential operators and the asymptotics of quantum mechanics, J. Funct. Anal. 29 (1978), 104-132.
J. L. Burchnall and T. W. Chaundy, Commutative ordinary differential operators, Proc. R. Soc. Lond. A 118 (1928), 557-583.
J. Richter, Burchnall-Chaundy theory for Ore extensions, 2013. arXiv:1309.4415v2 [math.RA].
V. Tulovsky, Some applications of Weyl calculus to Burchnall-Chaundy theory, I, Far East J. Math. Sci. (FJMS) 117(2) (2019), 113-118.
V. Tulovsky, Some applications of Weyl calculus to Burchnall-Chaundy theory, II, Far East J. Math. Sci. (FJMS) 138 (2022), 45-60.
B. L. van der Waerden, Algebra, Vol. I, Springer, 1991.
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