A sufficient nonsingularity condition for a discrete finite-gap one-energy two-dimensional Schrödinger operator on the quad-graph. (English. Russian original) Zbl 1330.81099
Funct. Anal. Appl. 49, No. 3, 210-213 (2015); translation from Funkts. Anal. Prilozh. 49, No. 3, 65-70 (2015).
Summary: The finite-gap approach is used to construct a two-dimensional discrete Schrödinger operator on a quad-graph, that is, a planar graph whose faces are quadrangles. The following definition of the nonsingularity of this operator is proposed: An operator is nonsingular if all of its coefficients are positive. Conditions on a spectral curve and a quad-graph sufficient for the operator constructed from them to be nonsingular are given.
MSC:
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 05C99 | Graph theory |
| 39A12 | Discrete version of topics in analysis |
| 14H70 | Relationships between algebraic curves and integrable systems |
| 14H81 | Relationships between algebraic curves and physics |
Keywords:
discrete operator; discrete complex analysis; finite-gap operator; spectral curve; M-curve; Riemann surface; nonsingularityReferences:
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