Topic
Spectrum of a matrix
About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.
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TL;DR: In this paper, a model operator H associated to a system of three particles on the threedimensional lattice ℤ3 that interact via nonlocal pair potentials is studied.
Abstract: A model operator H associated to a system of three particles on the threedimensional lattice ℤ3 that interact via nonlocal pair potentials is studied. The following results are established. (i) The operator H has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point if both the Friedrichs model operators
$$h_{\mu _\alpha } $$
(0), α = 1, 2, have threshold resonances. (ii) The operator H has finitely many eigenvalues lying outside the essential spectrum if at least one of the operators
$$h_{\mu _\alpha } $$
(0), α = 1, 2, has a threshold eigenvalue.
27 citations
TL;DR: The calculations exploit an analogy to the problem of finding a two-dimensional charge distribution on the interface of a semiconductor heterostructure under the influence of a split gate to find eigenvalues in a spectral interval of a large random matrix.
Abstract: We calculate the probability to find exactly $n$ eigenvalues in a spectral interval of a large random $N \times N$ matrix when this interval contains $s \ll N$ eigenvalues on average. The calculations exploit an analogy to the problem of finding a two-dimensional charge distribution on the interface of a semiconductor heterostructure under the influence of a split gate.
27 citations
TL;DR: In this article, the behavior of the determinant of the scattering matrix as a function on the spectrum of the unperturbed operator was studied for a large class of scattering systems, where the variation of this determinant was related to the number of eigenvalues due to perturbation.
27 citations
TL;DR: In this paper, the determinant of a sum of matrices was deduced from the theorem of Hamilton-Cayley, and a formula involving circular words and symmetric functions of the eigenvalues was given.
Abstract: We give a formula, involving circular words and symmetric functions of the eigenvalues, for the determinant of a sum of matrices. Theorem of Hamilton-Cayley is deduced from this formula.
27 citations
TL;DR: A matrix approximant to the generalized Frobenius-Perron equation is presented, the largest eigenvalues of which approach the most important eigen values of the operator (e.g., the so-called free energy).
Abstract: A matrix approximant to the generalized Frobenius-Perron equation is presented, the largest eigenvalues of which approach the most important eigenvalues of the operator (e.g., the so-called free energy). It is pointed out that also certain eigenfunctions not accessible to iteration of the continuous problem can easily be obtained in the discretized formalism. For general one-dimensional maps the spectrum of generalized entropies is shown to appear as the largest eigenvalue of a truncated version of the matrix. The statistical interpretation of the eigenfunction associated with the free energy is given in terms of configuration probabilities of spin chains.
27 citations