Bloch spectrum

Bloch spectrum

The Bloch spectrum is a concept in quantum mechanics.Let "H" be the one-dimensional Schrödinger equation operator

: H = - frac{d^2}{dx^2} + U_alpha,

where "Uα" is a periodic function of period "α". The Bloch spectrum ["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 ] 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 "E"0 < "E" from which certain closed intervals ["E"2"j"−1, "E"2"j"] ("j" = 1, 2, ...) are omitted. These are forbidden bands (or gaps) so the ("E"2"j"−2, "E"2"j"−1) are allowed bands.

References


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