Infinite Matrix Representations Of Isotropic Pseudodifferential Operators
METHODS AND APPLICATIONS OF ANALYSIS(2011)
摘要
We characterize isotropic pseudodifferential operators (elements of the Shubin calculus) by their action on Hermite functions. We show that a continuous linear operator A : S(R)-> S'(R) is an isotropic pseudodifferential operator of order r if and only if its "matrix" (K-(A)) m, n := L-2(R) is rapidly decreasing away from the diagonal {m = n}, order r/2 in m+n, where applying the discrete difference operator along the diagonal decreases the order by one. Here phi(m) is the m-th Hermite function. As an application, we give an isotropic version of the Beals commutator characterization of pseudodifferential operators, showing that if we define H-iso(s+r-2 beta)-(R) 2x + x2 to be the harmonic oscillator and Z the map extended linearly from Z(phi(k)) = phi(k) (1), then a continuous linear operator A : S(R). S'(R) is an isotropic pseudodifferential operator of order r if and only if commuting A alpha times with H and beta times with Z results in an bounded linear operator H-iso(s+r-2 beta)-(R)iso -> H-iso(s) (R), for all s is an element of R and alpha, beta. is an element of N-0.
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关键词
Pseudodifferential operators, harmonic oscillator, Shubin calculus, isotropic pseudodifferential operators, Beal's commutator characterization
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