## Abstract

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= - Δ + V(x) · , acting in the space L2(Rd)(d≥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(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}(Ω) is considered (Ω is an open domain in R^{d}).

Original language | English |
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Article number | 37 |

Journal | Integral Equations and Operator Theory |

Volume | 93 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2021 |

### Bibliographical note

Publisher Copyright:© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

## Keywords

- Discreteness of the spectrum
- Lagrangian relaxation
- Optimization problem
- Perturbations
- Rearrangement of a function
- Schrödinger operator

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory