2009/08/31 by Aleksey Kostenko, Mark Malamud · 1 citation
Mathematics · Physics and Astronomy · #math.SP #math-ph #math.MP #msc:34L05 #msc:34L40 #msc:47E05 #msc:47B25 #msc:47B36 #msc:81Q10
published as J. Differential Equations 249 (2010), 253-304 · 54 pages; several corrected typos, deleted Corollary 3.21, added references
arxiv created 2009/11/23 · arxiv updated 2010/05/17
Spectral properties of 1-D Schrödinger operators HX,α:=-(d2)/(d x2) + ∑_xn∈ Xαnδ(x-xn) with local point interactions on a discrete set X=\xn\n=1^∞ are well studied when d_*:=infn,k∈\N|xn-xk|>0. Our paper is devoted to the case d_*=0. We consider HX,α in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions. We show that the spectral properties of HX,α like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators HX,α to be self-adjoint, lower-semibounded, and discrete in the case d_*=0. The operators with δ'-type interactions are investigated too. The obtained results demonstrate that in the case d_*=0, as distinguished from the case d_*>0, the spectral properties of the operators with δ and δ'-type interactions are substantially different.