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Wannier functions for quasiperiodic finite-gap potentials. (English) Zbl 1178.14033
Theor. Math. Phys. 144, No. 2, 1081-1099 (2005); translation from Teor. Mat. Fiz. 144, No. 2, 234-256 (2005).
Summary: We consider Wannier functions of quasiperiodic \(g\)-gap (\(g\geq 1\)) potentials and investigate their main properties. In particular, we discuss the problem of averaging that underlies the definition of the Wannier functions for both periodic and quasiperiodic potentials and express Bloch functions and quasimomenta in terms of hyperelliptic \(\sigma \)-functions. Using this approach, we derive a power series for the Wannier function for quasiperiodic potentials valid for |x| \(\simeq 0\) and an asymptotic expansion valid at large distances. These functions are important in a number of applied problems.

MSC:
14H70 Relationships between algebraic curves and integrable systems
47E05 General theory of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
14H42 Theta functions and curves; Schottky problem
81V70 Many-body theory; quantum Hall effect
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