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This kind of function arises very naturally as the energy spectrum in what is called a quantum tight binding model. One considers a particle on some graph, say a square lattice, and defines a Hamiltonian by specifying an amplitude for the particle to hop from one site to another. If the lattice has a translation symmetry then one can diagonalize the Hamiltonian in terms of plane waves giving the function above with $t_{nm}$ the amplitude to hop n sites horizontally and m sites vertically. The shape of this energy spectrum is very important to the physics of fermions like electrons.