Strong unique continuation for the higher order fractional Laplacian

In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schr\"odinger operators. We deduce the strong unique continuation property in the presence of subcritical and critical Hardy type potentials. In the same setting, we address the unique continuation property from measurable sets of positive Lebesgue measure. As applications we prove the antilocality of the higher order fractional Laplacian and Runge type approximation theorems which have recently been exploited in the context of nonlocal Calder\'on type problems. As our main tools, we rely on the characterisation of the higher order fractional Laplacian through a generalised Caffarelli-Silvestre type extension problem and on adapted, iterated Carleman estimates.Comment: 50 page

Exceptional Legendre Polynomials and Confluent Darboux Transformations

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of "exceptional" degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.MAGF would like to thank the Max-Planck-Institute for Mathematics in the Sciences, Leipzig (Germany), where some of her work took place. DGU acknowledges support from the Spanish MICINN under grants PGC2018-096504-B-C33 and RTI2018-100754-B-I00 and the European Union under the 2014-2020 ERDF Operational Programme and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia (project FEDER-UCA18-108393)

Exceptional Gegenbauer polynomials via isospectral deformation

In this paper, we show how to construct exceptional orthogonal polynomials (XOP) using isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, where repeated factorizations at the same eigenvalue are allowed. These factorizations allow us to construct Sturm–Liouville problems with polynomial eigenfunctions that have an arbitrary number of realvalued parameters. We illustrate this new construction by exhibiting the class of deformed Gegenbauer polynomials, which are XOP families that are isospectral deformations of classical Gegenbauer polynomials.Spanish MINECO through Juan de la Cierva fellowship FJC2019-039681-I, Spanish State Research Agency through BCAM Severo Ochoa excellence accreditation SEV-2017-0718, Basque Government through the BERC Programme 2022-2025, projects PGC2018-096504-B-C33 and RTI2018-100754-B-I00 from FEDER/Ministerio de Ciencia e Innovacion-Agencia Estatal de Investigacion, the European Union under the 2014-2020 ERDF Operational Programme, and the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia (project FEDER-UCA18-108393

Exceptional Gegenbauer polynomials via isospectral deformation

In this paper, we show how to construct exceptional orthogonal polynomials (XOP) using isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, where repeated factorizations at the same eigenvalue are allowed. These factorizations allow us to construct Sturm-Liouville problems with polynomial eigenfunctions that have an arbitrary number of realvalued parameters. We illustrate this new construction by exhibiting the class of deformed Gegenbauer polynomials, which are XOP families that are isospectral deformations of classical Gegenbauer polynomials

A Bochner type characterization theorem for exceptional orthogonal polynomials

It was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be regarded as the extension of Bochner's result for classical orthogonal polynomials to the exceptional class. As a supplementary result, we derive a canonical form for exceptional operators based on a bilinear formalism, and prove that every exceptional operator has trivial monodromy at all primary poles

A Bochner type characterization theorem for exceptional orthogonal polynomials

It was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be regarded as the extension of Bochner's result for classical orthogonal polynomials to the exceptional class. As a supplementary result, we derive a canonical form for exceptional operators based on a bilinear formalism, and prove that every exceptional operator has trivial monodromy at all primary poles

Long-term results of the retrocapital metatarsal percutaneous osteotomy for hallux valgus

Producción CientíficaThe current trend in hallux valgus surgery is directed toward percutaneous procedures. However, no evidence that any of these methods of treatment are superior to the others has been described, excepting studies in the long term. The aim of this study was to analyse a series of patients who had undergone a percutaneous distal retrocapital osteotomy of the first metatarsal, and had been followed up for ten years. METHODS: We carried out a clinical and radiological evaluation of 115 feet ten years after surgery. RESULTS: The AOFAS scale results in the tenth postoperative year remained significantly favourable compared to their corresponding values in the preoperative period, yielding an improvement of 42.2 points overall on average. In relation to radiological findings, the mean hallux angle was maintained below 20 °, with a mean intermetatarsal angle of 8.1 °. CONCLUSION: Percutaneous retrocapital metatarsal osteotomy for treatment of mild to moderate hallux valgus is effective in the long term, with the advantages of a minimally invasive procedure