Absolute continuity of the spectrum of a two-dimensional Schrödinger operator with potential supported on a periodic system of curves. (English. Russian original) Zbl 0998.35006

St. Petersbg. Math. J. 12, No. 6, 983-1012 (2001); translation from Algebra Anal. 12, No. 6, 140-177 (2001).
Summary: The present paper is a continuation of M. Sh. Birman and T. A. Suslina [Two-dimensional periodic magnetic Hamiltonian is absolutely continuous, Algebra Anal. 9, No. 1, 32-48 (1997); translated in St. Petersbg. Math. J. 9, No. 1, 21-32 (1998; Zbl 0890.35096)]. Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential, Algebra Anal. 10, No. 4, 1-36 (1998); translated in St. Petersbg. Math. J. 10, No. 4, 579-601 (1999; Zbl 0922.35101] Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity, Algebra Anal. 11, No. 2, 1-40 (1999); translated in St. Petersbg. Math. J. 11, No. 2, 203-232 (2000; Zbl 0941.35015)]. A two-dimensional periodic magnetic Schrödinger operator with variable metric is considered. The electric potential is assumed to contain a term proportional to the \(\delta\)-function supported on a periodic system of piecewise-smooth curves. It is shown that, under rather general assumptions on the problem data, the spectrum of the Schrödinger operator is absolutely continuous.


35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs
35P05 General topics in linear spectral theory for PDEs