653 research outputs found
Magnetic interpretation of the nodal defect on graphs
In this note, we present a natural proof of a recent and surprising result of
Gregory Berkolaiko (arXiv 1110.5373) interpreting the "Courant nodal defect" of
a Schr\"odinger operator on a finite graph as a Morse index associated to the
deformations of the operator by switching on a magnetic field. This proof is
inspired by a nice paper of Miroslav Fiedler published in 1975
On the remainder in the Weyl formula for the Euclidean disk
We prove a 2-terms Weyl formula for the counting function N(mu) of the
spectrum of the Laplace operator in the Euclidean disk with a sharp remainder
estimate O(mu^2/3)
Semi-classical trace formulas and heat expansions
in the recent paper [Journal of Physics A, 43474-0288 (2011)], B. Helffer and
R. Purice compute the second term of a semi-classical trace formula for a
Schr\"odinger operator with magnetic field. We show how to recover their
formula by using the methods developped by the geometers in the seventies for
the heat expansions.Comment: To appear in "Analysis of Partial Differential Equations
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
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