On fractal structure of the spectrum for periodic point perturbations of the Schrödinger operator with a uniform magnetic field

Abstract

The band structure of the spectrum for a periodic Schrodinger operator H =-Δ + V in L 2(R v ) with a sufficiently nice potential V is a common thing in both the theoretical and mathematical physics [1]. On the other hand, using the direct integral decomposition over the torus T v one can give simple examples of periodic self-adjoint operators with a Cantor spectrum [2], but these operators are not local and therefore are not Schrodinger operators. (Recall that a closed operator A in L2(R v ) with domain D(A) is said to be local if for every φ∈D(A) the relation supp A φ⊂supp φ holds.) A curious phenomenon of a hidden fractal structure of the spectrum for periodic graph superlattices has been discovered in [3], this effect being no doubt of physical interest. In this connection, it is intriguing that there are local self-adjoint operators, namely the Schrodinger operators H with some periodic point potential, which have a Cantor spectrum on the semi-axis (—∞, 0) [4].