Perturbative expansions in  quantum mechanics

Mauricio D. Garay

Displaying similar documents to “Perturbative expansions in quantum mechanics”

The higher transvectants are redundant

Abdelmalek Abdesselam, Jaydeep Chipalkatti (2009)

Annales de l’institut Fourier

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Let $A, B$ denote generic binary forms, and let $𝔲_{r} = {(A, B)}_{r}$ denote their $r$ -th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the ${𝔲_{r}}$ . As a consequence, we show that each of the higher transvectants ${𝔲_{r} : r \geq 2}$ is redundant in the sense that it can be completely recovered from $𝔲_{0}$ and $𝔲_{1}$ . This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of $S L_{2}$ -representations,...

Elementary linear algebra for advanced spectral problems

Johannes Sjöstrand, Maciej Zworski (2007)

Annales de l’institut Fourier

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We describe a simple linear algebra idea which has been used in different branches of mathematics such as bifurcation theory, partial differential equations and numerical analysis. Under the name of the Schur complement method it is one of the standard tools of applied linear algebra. In PDE and spectral analysis it is sometimes called the Grushin problem method, and here we concentrate on its uses in the study of infinite dimensional problems, coming from partial differential operators...

An explicit formula for the Hilbert symbol of a formal group

Floric Tavares Ribeiro (2011)

Annales de l’institut Fourier

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A Brückner-Vostokov formula for the Hilbert symbol of a formal group was established by Abrashkin under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of ( $ϕ, Γ$ )-modules and a cohomological interpretation of Abrashkin’s technique. To do this, we build ( $ϕ, Γ$ )-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas...

Hasse–Schmidt derivations, divided powers and differential smoothness

Luis Narváez Macarro (2009)

Annales de l’institut Fourier

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Let $k$ be a commutative ring, $A$ a commutative $k$ -algebra and $D$ the filtered ring of $k$ -linear differential operators of $A$ . We prove that: (1) The graded ring $gr D$ admits a canonical embedding $θ$ into the graded dual of the symmetric algebra of the module $Ω_{A / k}$ of differentials of $A$ over $k$ , which has a canonical divided power structure. (2) There is a canonical morphism $ϑ$ from the divided power algebra of the module of $k$ -linear Hasse–Schmidt integrable derivations of $A$ to $gr D$ . (3) Morphisms $θ$ and...

Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups

Cédric Bonnafé, Christophe Hohlweg (2006)

Annales de l’institut Fourier

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We construct a subalgebra $Σ^{'} (W_{n})$ of dimension $2 \cdot 3^{n - 1}$ of the group algebra of the Weyl group $W_{n}$ of type $B_{n}$ containing its usual Solomon algebra and the one of $𝔖_{n}$ : $Σ^{'} (W_{n})$ is nothing but the Mantaci-Reutenauer algebra but our point of view leads us to a construction of a surjective morphism of algebras $Σ^{'} (W_{n}) \to Z Irr (W_{n})$ . Jöllenbeck’s construction of irreducible characters of the symmetric group by using the coplactic equivalence classes can then be transposed to $W_{n}$ . In an appendix, P. Baumann and C. Hohlweg present in an...