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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 Schrodinger 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.
According to a well-know theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed that one half of Sturms theorem for the strings applies to the theory of membranes: N-th eigenf
It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the distribution of the n
A foundational result in the theory of Lyndon words (words that are strictly earlier in lexicographic order than their cyclic permutations) is the Chen-Fox-Lyndon theorem which states that every word has a unique non-increasing decomposition into Lyn
In this work we present a formalism of abstract quantum field theory for fat graphs and its realizations. This is a generalization of an earlier work for stable graphs. We define the abstract correlators $mathcal F_g^mu$, abstract free energy $mathca
We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros $phi$ of the $n$-th eigenfunction of the Schrodinger operator on a quantum graph is related to the stability of the $n$-th eigenvalue of the perturbation of the