This work is a continuation of our previous paper (Zelenko in Appl Anal Optim 3(2):281–306, 2019), where for the Schrödinger operator \(H=-\Delta + V({{\mathbf {x}}})\cdot \), acting in the space \(L_2({{\mathbf {R}}}^d)\,(d\ge 3)\), some sufficient conditions for discreteness of its spectrum have been obtained on the base of well known Mazya–Shubin criterion and an optimization problem for a set function. This problem is an infinite-dimensional generalization of a binary linear programming problem. A sufficient condition for discreteness of the spectrum is formulated in terms of the non-increasing rearrangement of the potential \(V({{\mathbf {x}}})\). Using the method of Lagrangian relaxation for this optimization problem, we obtain a sufficient condition for discreteness of the spectrum in terms of expectation and deviation of the potential. By means of suitable perturbations of the potential we obtain conditions for discreteness of the spectrum, covering potentials which tend to infinity only on subsets of cubes, whose Lebesgue measures tend to zero when the cubes go to infinity. Also the case where the operator H is defined in the space \(L_2(\Omega )\) is considered (\(\Omega \) is an open domain in \({{\mathbf {R}}}^d\)).

这项工作是我们之前论文（Zelenko in Appl Anal Optim 3(2):281–306, 2019）的延续，其中对于薛定谔算子\(H=-\Delta + V({{\mathbf {x}} })\cdot \)，作用于空间\(L_2({{\mathbf {R}}}^d)\,(d\ge 3)\)，在众所周知的 Mazya-Shubin 准则和集合函数的优化问题的基础。这个问题是二元线性规划问题的无限维推广。谱离散性的充分条件根据势能\(V({{\mathbf {x}}})\). 对这个优化问题使用拉格朗日松弛方法，我们得到了谱离散的充分条件，即期望和电位偏差。通过适当的电势扰动，我们获得了频谱离散性的条件，覆盖了仅在立方体的子集上趋于无穷大的电势，当立方体趋于无穷大时，其勒贝格测度趋于零。还考虑在空间\(L_2(\Omega )\) 中定义算子H的情况（\(\Omega \)是\({{\mathbf {R}}}^d\) 中的一个开放域））。