Bloch spectrum

The Bloch spectrum is a concept in quantum mechanics in the field of theoretical physics; this concept addresses certain energy spectra considerations. Let H be the one-dimensional Schrödinger equation operator

${\displaystyle H=-{\frac {d^{2}}{dx^{2}}}+U_{\alpha },}$

where U_α is a periodic function of period α. The Bloch spectrum^[1] of H is defined as the set of values E for which all the solutions of (H − E)φ = 0 are bounded on the whole real axis. The Bloch spectrum consists of the half-line E₀ < E from which certain closed intervals [E_2j−1, E_2j] (j = 1, 2, ...) are omitted. These are forbidden bands (or gaps) so the (E_2j−2, E_2j−1) are allowed bands.

References

1. "An upper bound on the allowed bands of the Bloch spectrum of one-dimensional Schrödinger operators with periodic potentials", N. N. Meiman, Journal of Mathematical Physics, March, 1983, volume 24, issue 3, pp. 539–540