An elementary treatise on determinants by Charles L. Dodgson

By Charles L. Dodgson

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Quantum and Arithmetical Chaos 49 II. Statistical Distribution of Quantum Eigenvalues Wigner and Dyson in the fifties had proposed to describe complicated (and mostly unknown) Hamiltonian of heavy nuclei by a member of an ensemble of random matrices and they argued that the type of this ensemble depends only on the symmetry of the Hamiltonian. For systems without time-reversal invariance the relevant ensemble is the Gaussian Unitary Ensemble (GUE), for systems invariant with respect to time-reversal the ensemble is the Gaussian Orthogonal Ensemble (GOE) and for systems with time-reversal invariance but with half-integer spin energy levels have to be described according to the Gaussian Symplectic Ensemble (GSE) of random matrices.

The next order equation 2∇S∇A + ∆SA = 0 38 Eugene Bogomolny is equivalent to the conservation of current. Indeed, for the semiclassical wave function (35) 1 J = (Ψ ∗ ∇Ψ − Ψ ∇Ψ ∗ ) = A2 ∇S 2i and ∇J = A(2∇A∇S + A∆S) = 0 . The solution of the above transport equation has the form A(x, y) = ∂2S π 1 det − ∂ti⊥ ∂tf ⊥ (2π )(f +1)/2 ki kf 1/2 where ti⊥ and tf ⊥ are coordinates perpendicular to the trajectory in the initial, y, and final, x, points respectively and ki , kf are the initial and final momenta.

6. Small and large action contributions to the Green function for nearby points y→x GE (x, y) −→ eip(x−y)/ dp . f (2π ) (E − H(p, x) + i ) Therefore Im GE (x, x) = −π dp δ (E − H(p, x)) (2π )f and the smooth part of the level density in the leading approximation equals the phase-space volume of the constant energy surface divided by (2π )f ¯ d(E) = dpdx δ (E − H(p, x)) . (2π )f The contribution from long classical trajectories with finite actions corresponds to the oscillating part of the density and can be calculated using the semiclassical approximation of the Green function (35).

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