André Unterberger's Alternative Pseudodifferential Analysis: With an Application PDF

By André Unterberger

ISBN-10: 3540779108

ISBN-13: 9783540779100

ISBN-10: 3540779116

ISBN-13: 9783540779117

This quantity introduces a wholly new pseudodifferential research at the line, the competition of which to the standard (Weyl-type) research might be stated to mirror that, in illustration conception, among the representations from the discrete and from the (full, non-unitary) sequence, or that among modular types of the holomorphic and alternative for the standard Moyal-type brackets. This pseudodifferential research is determined by the one-dimensional case of the lately brought anaplectic illustration and research, a competitor of the metaplectic illustration and ordinary analysis.

Besides researchers and graduate scholars attracted to pseudodifferential research and in modular varieties, the publication can also attract analysts and physicists, for its suggestions making attainable the transformation of creation-annihilation operators into automorphisms, at the same time altering the standard scalar product into an indefinite yet nonetheless non-degenerate one.

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Extra resources for Alternative Pseudodifferential Analysis: With an Application to Modular Forms

Example text

1, one obtains, given j, k ∈ Z, that k ∑ (φζj | Opasc m (hm ) φζ ) m≥1 j−k−1 = ∑ m=1 m m+1 π 2 4π Π (Θm hm )(z) (φζj | A−m−1 φζk ) (Im z)m+1 d µ (z) z j−k−1 = ∑ m=1 Π m+1 m m+1 j,k π 2 Cm (Im ζ ) 2 × 4π −m−1+ j−k −m−1− j+k 2 2 (Θm hm )(z) (¯z − ζ¯ ) (¯z − ζ ) (Im z)m+1 d µ (z). 34). It is easy to deal with linear operators: Eζ → Eζ because, within this class, one can always multiply any operator, on the left or on the right, by any linear combination of Q and P. 34). 6 should continue to hold. 4 is still valid.

X , B] . . 51) s−1∈S( j−k−1) j,k j,k where the functions TXj,k ,s are holomorphic. When = 0, we abbreviate TX ,s as Ts . asc The map Op is one to one. 1 for every s. ↑ Proof. 51) to start with. Set χm+1 = Θm hm (cf. 1) for simplicity of notation. 1 that only the terms such that m ∈ S( j − k − 1) (cf. 50)) can contribute to this scalar product. 1). 51) of this function of ζ as a polynomial in (Im ζ ) 2 with holomorphic coefficients if one sets s = n + 1 + 2r, a number in the finite set characterized by the condition that s − 1 ∈ S( j − k − 1): then, for any given s, the domain of possible n’s is the set S(s − 1).

X , B] . . ]]φζk ) is zero unless j − k ≥ + m0 + 1. 6) in a way including this extra piece of information: it will help in the proof by induction of that theorem. ↑ (iii) In particular, (φζj | Opasc (h) φζk ) = 0 for every h ∈ (Sweak (R2 )) and every ζ ∈ Π unless j −k ≥ 2. 11) the operator A∗z = π 2 (Q − z P) for Az = π 2 (Q − z¯ P). In the conjugate calculus, the above condition changes to k − j ≥ 2. 3, explains the link between the two calculi. As will be seen in Sect. 4, a composition formula exists for the first (hence for each) of the two symbolic calculi.

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Alternative Pseudodifferential Analysis: With an Application to Modular Forms by André Unterberger


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