21 research outputs found
Nodal count of graph eigenfunctions via magnetic perturbation
We establish a connection between the stability of an eigenvalue under a
magnetic perturbation and the number of zeros of the corresponding
eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a
graph and count the number of edges where the eigenfunction changes sign (has a
"zero"). It is known that the -th eigenfunction has such zeros,
where the "nodal surplus" is an integer between 0 and the number of cycles
on the graph.
We then perturb the Laplacian by a weak magnetic field and view the -th
eigenvalue as a function of the perturbation. It is shown that this function
has a critical point at the zero field and that the Morse index of the critical
point is equal to the nodal surplus of the -th eigenfunction of the
unperturbed graph.Comment: 18 pages, 4 figure
Eigenvalues, absolute continuity and localizations for periodic unitary transition operators
The localization phenomenon for periodic unitary transition operators on a
Hilbert space consisting of square summable functions on an integer lattice
with values in a complex vector space, which is a generalization of the
discrete-time quantum walks with constant coin matrices, are discussed. It is
proved that a periodic unitary transition operator has an eigenvalue if and
only if the corresponding unitary matrix-valued function on a torus has an
eigenvalue which does not depend on the points on the torus. It is also proved
that the continuous spectrum of the periodic unitary transition operators is
absolutely continuous. As a result, it is shown that the localization happens
if and only if there exists an eigenvalue, and the long time average of the
transition probabilities coincides with the point-wise norm of the projection
of the initial state to the direct sum of eigenspaces.Comment: 15 pages. The first half of the previous version has been simplified.
Some of previous results has been mentioned. As an example, Grover walk on a
topological crystal is explaine