Non-invertibility of certain almost Mathieu operators

Balasubramanian, R.; Kulkarni, S.; Radha, R.

doi:10.1090/S0002-9939-00-05760-9

Non-invertibility of certain almost Mathieu operators
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by R. Balasubramanian, S. H. Kulkarni and R. Radha PDF

Proc. Amer. Math. Soc. 129 (2001), 2017-2018 Request permission

Abstract:

It is shown that the almost Mathieu operators of the type $Te_n~=$ $e_{n-1}+\lambda sin(2nr)e_n+e_{n+1}$ where $\lambda$ is real and $r$ is a rational multiple of $\pi$ and $\{e_n:n=1,2,3,...\},$ an orthonormal basis for a Hilbert space, is not invertible.

References

J. Avron, P. H. M. van Mouche, and B. Simon, On the measure of the spectrum for the almost Mathieu operator, Comm. Math. Phys. 132 (1990), no. 1, 103–118. MR 1069202
J. Béllissard and B. Simon, Cantor spectrum for the almost Mathieu equation, J. Functional Analysis 48 (1982), no. 3, 408–419. MR 678179, DOI 10.1016/0022-1236(82)90094-5
Svetlana Ya. Jitomirskaya and Yoram Last, Anderson localization for the almost Mathieu equation. III. Semi-uniform localization, continuity of gaps, and measure of the spectrum, Comm. Math. Phys. 195 (1998), no. 1, 1–14. MR 1637389, DOI 10.1007/s002200050376
Y. Last, Almost everything about the almost Mathieu operator. I, XIth International Congress of Mathematical Physics (Paris, 1994) Int. Press, Cambridge, MA, 1995, pp. 366–372. MR 1370693
Y. Last, Zero measure spectrum for the almost Mathieu operator, Comm. Math. Phys. 164 (1994), no. 2, 421–432. MR 1289331