Spectral Properties of Schrödinger Operators on Lattices
Summary
The spectral analysis of Schrödinger operators on discrete lattices investigates how energy levels and wave functions distribute when quantum particles hop between lattice sites under the influence of a potential. Central to this field is the distinction between the essential spectrum, which describes extended states and continuous energy bands, and the discrete spectrum, comprising isolated eigenvalues associated with bound states. Perturbative techniques, often based on convolution‐type hopping operators or rank-one potentials, reveal how small changes in coupling or geometry induce threshold effects, generate new eigenvalues near band edges and modify localisation properties. These insights underpin applications in condensed-matter physics, such as transport in optical lattices, edge states in topological materials and stability phenomena in ultracold atomic gases.
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Spectral Properties of Schrödinger Operators on Lattices publication trend
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Technical terms
Essential spectrum: The set of energy values corresponding to delocalised or extended states, typically forming continuous bands.
Discrete spectrum: Isolated eigenvalues of finite multiplicity associated with spatially localised bound states.
Lattice: A regular grid of discrete points in one or more dimensions on which particles hop.
Schrödinger operator: A Hamiltonian describing kinetic hopping between sites and potential energy at each site.
Eigenvalue: A specific energy at which the operator admits a non-trivial solution (eigenfunction) satisfying the Schrödinger equation.
Convolution operator: A translation-invariant kinetic term defined by a hopping kernel on the lattice.
Threshold effect: The phenomenon whereby eigenvalues appear or disappear as a coupling parameter crosses a critical value near the band edge.
References
- Spectrum of One-Dimensional Potential Perturbed by a Small Convolution Operator: General Structure. Mathematics (2023).
- On the spectrum of Schrödinger-type operators on two dimensional lattices. Journal of Mathematical Analysis and Applications (2022).
- Expansion of eigenvalues of the perturbed discrete bilaplacian. Monatshefte für Mathematik (2022).
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