Schrödinger operators : proceedings of the Nordic Summer by H Holden; A Jensen

By H Holden; A Jensen

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Extra info for Schrödinger operators : proceedings of the Nordic Summer School in Mathematics held at Sandbjerg Slot, Sønderborg, Denmark, August 1-12, 1988

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A basic property of linear relations between symplectic vector spaces is the following symplectic functorial rule. 1. 4) A proof of this fundamental formula is given in Sect. 4 By using this formula we can easily prove the following two fundamental statements. 2. 5 Proof. If R§ = R and S § = S, then (S ◦ R)§ = S § ◦ R§ = S ◦ R. 3. The image R ◦ K of an isotropic (coisotropic, Lagrangian) subspace K by a linear symplectic relation R is an isotropic (coisotropic, Lagrangian) subspace. 4 It is taken from (Benenti 1988) and it is rather cumbersome.

V ω(u, v) = 0 ι I u K isotropic (M, ω) ι ω=0 Fig. 2. 12) The left-hand side of this equation is the precursor of the so-called Lagrange bracket. ♦ This suggests the following extension of the definition of Lagrangian submanifold: a Lagrangian immersion is an immersion ι : K → M into a symplectic manifold such that dim K = 12 dim M and ι∗ ω = 0. An immersed Lagrangian submanifold is the image Λ = ι(K) of a Lagrangian immersion.

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