2020/11/30 by Leonid Zelenko, Zelenko, Leonid
Mathematics · #35P05 #47B25 #90C10 #90C27 #91A12 #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems #Primary 47F05 #Secondary 81Q10 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #\\47D08
paper · pdf · doi:10.48550/arxiv.2011.14676
openalex publication_date 2020/11/30 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
This work is a continuation of our previos paper, where for the Schr "odinger\noperator H=-\Δ+ V( e)\⋅ (V( e)\≥ 0), acting in the space\nL2( Rd) ,(d\≥ 3), some sufficient conditions for discreteness of its\nspectrum have been obtained on the base of well known Mazya -Shubin criterion\nand an optimization problem for a set function, which is an\ninfinite-dimensional generalization of a binary linear programming problem. A\nsufficient condition for discreteness of the spectrum is formulated in terms of\nthe non-increasing rearrangement of the potential V( e). Using the method of\nLagrangian relaxation for this optimization problem, we obtain a sufficient\ncondition for discreteness of the spectrum in terms of expectation and\ndeviation of the potential. By means of suitable perturbations of the potential\nwe obtain conditions for discreteness of the spectrum, covering potentials\nwhich tend to infinity only on subsets of cubes, whose Lebesgue measures tend\nto zero when the cubes go to infinity. Also the case where the operator H is\ndefined in the space L2(\Ω) is considered (\Ω is an open domain in\n Rd).\n