47A55 Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
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Keywords
- dissipative operators (2)
- generic perturbation (2)
- perturbation analysis (2)
- rank one perturbation (2)
- Abel basis of root vectors (1)
- Brunovsky form (1)
- Feshbach decomposition (1)
- Lax-Phillips scattering theory (1)
- Spectra of Graphs, Laplacians, Perturbations of Eigenvalues (1)
- carrier and current densities (1)
This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity) and its associated eigenvector, the so-called Fiedler vector. Here we prove that, given a Laplacian matrix, it is possible to perturb the weights of the existing edges in the underlying graph in order to obtain simple eigenvalues and a Fiedler vector composed of only non-zero entries. These structural genericity properties with the constraint of not adding edges in the underlying
graph are stronger than the classical ones, for which arbitrary structural perturbations are allowed. These results open the opportunity to understand the impact of structural changes on the dynamics of complex systems.
We study the perturbation theory of structured matrices under
structured rank one perturbations, and then focus on several classes of complex
matrices. Generic Jordan structures of perturbed matrices are identified.
It is shown that the perturbation
behavior of the Jordan structures
is substantially different from the corresponding theory for unstructured generic
rank one perturbations.
For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in
the sense of algebraic geometry. Special attention is paid to the perturbation
behavior of the sign characteristic. Typically, under such a perturbation,
for every given eigenvalue, the largest Jordan block of the eigenvalue is
destroyed and (in case the eigenvalue is real) all other Jordan blocks
keep their sign characteristic. The new eigenvalues, i.e., those eigenvalues of
the perturbed matrix that are not eigenvalues of the original matrix,
are typically simple, and in some cases information is provided about their sign
characteristic (if the new eigenvalue is real). The main results are proved by using
the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky
canonical form and on general results concerning rank one perturbations.
For an operator-valued block-matrix model, which is called in quantum physics a Feshbach decomposition, a scattering theory is considered. Under trace class perturbation the channel scattering matrices are calculated. Using Feshbach's optical potential it is shown that for a given spectral parameter the channel scattering matrices can be recovered either from a dissipative or from a Lax-Phillips scattering theory.
We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schroedinger-type operator, the spectral properties of which will be investigated.
Non-selfadjoint operators play an important role in the modeling of open quantum systems. We consider a one-dimensional Schroedinger-type operator with dissipative boundary conditions and dissipative delta potentials. An explicit description of the characteristic function, the minimal dilation and the generalized eigenfunctions of the dilation is given. The quantities of carrier and current densities are rigorously defined. Furthermore we will show that the current is not constant and that the variation of the current depend essentially on the chosen density matrix and imaginary parts of the delta potentials. This correspondence can be used to model a recombination-generation rate in the open quantum system.