By Joachim Asch, Alain Joye

At the QMath9 assembly, younger scientists know about the state-of-the-art within the mathematical physics of quantum platforms. according to that occasion, this booklet deals a range of exceptional articles written in pedagogical sort comprising six sections which disguise new innovations and up to date effects on spectral idea, statistical mechanics, Bose-Einstein condensation, random operators, magnetic Schrödinger operators and lots more and plenty extra. For postgraduate scholars, Mathematical Physics of Quantum Systems serves as an invaluable creation to the examine literature. For extra specialist researchers, this ebook may be a concise and glossy resource of reference.

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It is well known that to prove localization of Hλ,α,θ it suffices to prove that all polynomially bounded solutions of Hλ,α,θ Ψ = EΨ decay exponentially. We will use the notation G[x1 ,x2 ] (x, y) for matrix elements of the Green’s function (H − E)−1 of the operator Hλ,α,θ restricted to the interval [x1 , x2 ] with zero boundary conditions at x1 − 1 and x2 + 1. It can be checked easily that values of any formal solution Ψ of the equation HΨ = EΨ at a point x ∈ I = [x1 , x2 ] ⊂ Z can be reconstructed from the boundary values via Ψ (x) = −GI (x, x1 )Ψ (x1 − 1) − GI (x, x2 )Ψ (x2 + 1) .

E ln((r + δ)/r) ((|τ | − r − δ)/4 − 2a2 )3/2 Thus τ → θn ({τ },1,r) is integrable at infinity for any r > 0, and in particular for r = τc + 2δ. According to (28), we have therefore shown the classical Landau-Zener formula |φn−ε (∞)|2 = e− a2 π ε + O(ε 6 e− 1 a2 π ε ). For the Landau-Zener model, the transition probabilities can also be proved by the method of [7]. e. the level lines Im(τ (t)) = Im(τ (tc )), play an essential role. In particular, the method requires that an anti-Stokes line emanating from the critical point tc of τ stays in a strip of finite width around the real axis as Re(t) → ±∞.

99, 225–246 (1990). 12. L. H. Eliasson, Floquet solutions for the one-dimensional quasi-periodic Schr¨ odinger equation. Commun. Math. Phys. 146, 447–482 (1992). 13. S. Jitomirskaya, Metal-Insulator transition for the almost Mathieu operator. Annals of Math. 150, 1159–1175 (1999). 14. S. Ya. Jitomirskaya, I. V. Krasovsky, Continuity of the measure of the spectrum for discrete quasiperiodic operators. Math. Res. Lett. 9, no. 4, 413–421 (2002). 15. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc.

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