By Laurent Schwartz

ISBN-10: 0677300905

ISBN-13: 9780677300900

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The contribution of a single-particle energy level En( ) to the persistent current is given by p dEn/d , where the mean density state ρ is the inverse of mean level separation. The flux will now be scaled by flux quanta =hc/e . 22) where ' integration and < >n imply the averages for one period and for levels lying below the Fermi level respectively. Fig. 2. Aharonov-Bohm flux and Sinai billiard with persistent current. 23), the suffix signifies the additional procedure of coarsegraining the fine structures less than the mean level spacing ( =(2 p)-1) ; is just the density of states with its average given by ρ .

1988). Rev. Mod. Phys. 60, 537. Chapter 4 Chaos and Quantum Transport in Open Magnetic Billiards: from Stadium to Sinai Billiards Particle motion inside or outside billiards provides a prototype of chaos in conservative dynamical systems. In this chapter, chaos and quantum transport in magnetic billiards are investigated. First, theoretical and experimental studies are presented on quantum transport in weakly-opened circle and stadium billiards in the perpendicular magnetic field B. While in the circle, the magnetoconductance shows grossly regular oscillations, in the stadium it exhibits a transition from the mild to violent undulations with increase of B.

Although no positive Lyapunov exponent is available, it shows a stagnant chaos around a torus in the weak field case, while, in the strong field case, wandering from one stagnant region to another in an unpredictable way. The power spectra of transmission coefficients show a universal 1/f 2 behavior, which will be pointed out to be caused by a breakdown of the law of large numbers. Second, we shall investigate pseudo-chaos in multiple resonant tunnelings through a double-barrier structure. Both phenomena will be realized by using GaAs/AlAs heterostructures.

### Application of distributions to the theory of elementary particles in quantum mechanics by Laurent Schwartz

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