2018/12/02 by Leonid Zelenko, Zelenko, Leonid
Mathematics · #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1812.00416
openalex publication_date 2018/12/02 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28
For the the Schr "odinger operator H=-\Δ+ V(x)\⋅, acting in the\nspace L2( Rd) ,(d\≥ 3), with V(x)\≥ 0 and V(\⋅)\∈ L1,loc( Rd), we\nobtain some constructive conditions for discreteness of its spectrum. Basing on\nthe Mazya-Shubin criterion for discreteness of the spectrum of H and using\nthe isocapacity inequality and the concept of base polyhedron for the harmonic\ncapacity, we have estimated from below the cost functional of an optimization\nproblem, involved in this criterion, replacing a submodular constrain (in terms\nof the harmonic capacity) by a weaker but additive constrain (in terms of a\nmeasure). By this way we obtain an optimization problem, which can be\nconsidered as an infinite-dimensional analogue of the optimal covering problem.\nWe have solved this problem for the case of a non-atomic measure. This approach\nenables us to obtain for the operator H some sufficient conditions for\ndiscreteness of its spectrum in terms of non-increasing rearrangements, with\nrespect to measures from the base polyhedron, for some functions connected with\nthe potential V(x). We construct some counterexamples, which permit to compare\nour results between themselves and with results of other authors.\n