Characteristic Vectors of Bordered Matrices With Infinite Dimensions

Quantum chaos in many-body systems: what can we learn from the Ce atom?

“…The BW shape of the strength function was first derived for the infinite-size band random matrix (BRM) model [37,38]. It was assumed in the model that the diagonal matrix elements are equally spaced H kk = kD (thus, ρ = D −1 = const), and the off-diagonal matrix elements are independent random variables with H ij = 0, and H 2 ij = V 2 for |i − j | ≤ b (b 1 characterizes the width of the band) and H ij = 0 outside the band.…”

“…It is caused by an effective bandedness of the Hamiltonian matrix, which means that for greater |E −E k | the coupling decreases, and the mixing is achieved effectively through higher perturbation theory orders. This becomes especially obvious in the Wigner BRM model where the decrease of the strength function outside the band is exponential [8,37,38].…”

Correlation of financial markets in times of crisis

“…One tool that was first developed in nuclear physics for studying complex systems with unknown correlation structure is random matrix theory [67]- [70], which confronts the results obtained for the eigenvalues of the correlation matrix of a real system with those of the correlation matrix obtained from a pure random matrix. This approach was successfully applied to a large number of financial markets [71]- [100], and also to the relation between world markets [101]- [102].…”

Noncommutative probability in classical disordered systems

“…In the large N limit, by the Wigner theorem, see [19]- [22], the system of p random matrices with independent variables will give rise to the quantum Boltzmann algebra with p degrees of freedom with the generators A a , A † a , a = 0, . .…”