Mathematical Heritage of Sergey Naboko: Functional Models of Non-Self-Adjoint Operators

This is an overview of mathematical heritage of Sergey Naboko in the area of functional models of non-self-adjoint operators. It covers the works by Sergey in model construction, the analysis of absolutely continuous and singular spectra and the construction of the scattering theory in model terms.Comment: 14 pages. arXiv admin note: text overlap with arXiv:2204.0119

Asymptotic analysis of operator families and applications to resonant media

We give an overview of operator-theoretic tools that have recently proved useful in the analysis of boundary-value and transmission problems for second-order partial differential equations, with a view to addressing, in particular, the asymptotic behaviour of resolvents of physically motivated parameter-dependent operator families. We demonstrate the links of this rich area, on the one hand, to functional frameworks developed by S. N. Naboko and his students, and on the other hand, to concrete applications of current interest in the physics and engineering communities.Comment: 60 pages, 2 figures; a survey of recent results in the area, see also arXiv:2010.13318, arXiv:1808.03961, arXiv:1703.06220, arXiv:1510.0336

Сase of successful treatment of chronic gastric remnant haemorrhagic ulcer after Roux-en-Y gastric bypass

Acute bleeding from a gastric remnant ulcer is an exceptionally rare complication, especially if it occurs several years after bariatric surgery. We present a case of a patient with a bleeding gastric remnant ulcer that occurred 6 years (2013–2019) after the Roux-en-Y Gastric Bypass (RYGB) performed for morbid obesity. The patient was urgently hospitalized in the surgical department with the clinic of gastrointestinal bleeding, according to fibroesophagogastroscopy, no sources of bleeding were found in the examined departments. On the background of conservative treatment, there was an improvement in the condition, but on the fifth day of hospitalization there was a clinic of recurrence of bleeding, while repeated emergency endoscopic examination of the sources also did not reveal. In this regard, it was decided to perform surgery to examine the «residual» stomach. With the help of laparoscopic transgastric gastroduodenoscopy, we were able to detect the source of active bleeding in the stomach excluded from digestion - a bleeding ulcer in the area of small curvature. The removal of this stomach was performed with a good postoperative recovery: the patient was discharged in a satisfactory condition for outpatient treatment, currently feels well, returned to work

Schrödinger operators with δ and δ′-potentials supported on hypersurfaces

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements