Fixed angle scattering: Recovery of singularities and its limitations

We prove that in dimension n ≥ 2 the main singularities of a complex potential q having a certain a priori regularity are contained in the Born approximation qθ constructed from fixed angle scattering data. Moreover, q-qθ can be up to one derivative more regular than q in the Sobolev scale. In fact, this result is optimal. We construct a family of compactly supported and radial potentials for which it is not possible to have more than one derivative gain. Also, these functions show that for n > 3, the maximum derivative gain can be very small for potentials in the Sobolev scale not having a certain a priori level of regularity which grows with the dimension.The author was supported by Spanish government predoctoral grant BES-2015- 074055 (project MTM2014-57769-C3-1-P

Resolvent estimates for the magnetic Schr\"odinger operator in dimension n≥2n \geq 2

It is well known that the resolvent of the free Schr\"odinger operator on weighted $L^2$ spaces has norm decaying like $\lambda^{-\frac{1}{2}}$ at energy $\lambda$. There are several works proving analogous high-frequency estimates for magnetic Schr\"odinger operators, with large long or short range potentials, in dimensions $n \geq 3$. We prove that the same estimates remain valid in all dimensions $n \geq 2$.Comment: 21 page

Recovery of singularities in inverse scattering

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 14-12-201

Uniqueness for the inverse fixed angle scattering problem

We present a uniqueness result in dimensions $2$ and $3$ for the inverse fixed angle scattering problem associated to the Schr\"odinger operator $-\Delta+q$, where $q$ is a small real valued potential with compact support in the Sobolev space $W^{\beta,2}$ with $\beta>0.$ This result improves the known result, due to Stefanov, in the sense that almost no regularity is required for the potential. The uniqueness result still holds in dimension $4$, but for more regular potentials in $W^{\beta,2}$ with $\beta>2/3$

Exploring the perception of barriers to a dual career by student-athletes with/out disabilities

n recent years, there has been an increase in knowledge about the barriers experienced by people with disabilities in the education system or sports. However, no studies have analyzed the barriers for those who try to succeed in both disciplines (dual career). The purpose of this study was to examine the barriers faced by student-athletes with/out disability to a dual career combining studies and sport. Two groups were involved in the study (n = 162): student-athletes with disabilities (n = 79) and student-athletes without disabilities (n = 83). Data collected included: (a) socio-demographic aspects; and (b) barriers towards achieving a good balance between sport and academics during the dual career, through the "Perceptions of dual career student-athletes" (ESTPORT) questionnaire. The results showed that student-athletes with disabilities were more likely to perceive in a greater extent the barriers, the university is far from my home (p = 0.007) and the university is far from my training site (p = 0.006), I find myself unable to balance study and training time (p = 0.030), I have to take care of my family (p<0.001), and my current job does not allow me to study enough (p<0.001). The MANOVA analysis showed that the factors gender, competitive level, and employment status had an influence on the perception of some barriers between groups. In conclusion, student-athletes with disabilities perceived barriers more strongly than those without disabilities, and measures are needed to ensure their inclusion in the education system

Influence of an Educational Innovation Program and Digitally Supported Tasks on Psychological Aspects, Motivational Climate, and Academic Performance

Background: In university education, there is a need to provide students with the ability to use knowledge, and it has been shown that the cooperative model, with respect to information and communication technology (ICT), is effective. The aim of this study was to analyze the influence of an educational innovation program, based on the jigsaw technique and digitally supported tasks, on the psychological aspects, motivational climate, and academic performance of university students. Methods: A quasi-experimental study was conducted with an experimental group consisting of 100 university students (mean age: 21.84 ± 1.50 years). The motivational climate and the basic psychological needs in education, intrinsic motivation, academic self-concept, and academic performance were measured. Results: Significant increases were found in all variables after the intervention (p < 0.006–0.001), except for the variable, ego-motivational climate. The covariate perception of prior competences was significant for the model (p < 0.001). The students who had chosen a specific topic to develop with the jigsaw technique obtained a better grade than the rest of their classmates when the student’s academic performance was included as a covariate (p < 0.001). Conclusions: The psychological aspects, motivational climates, and academic performances of university students improved after the implementation of an educational innovation program, based on the cooperative learning model with the jigsaw technique, and the use of digitally supported tasks.Educació

The Fixed Angle Scattering Problem with a First-Order Perturbation

We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by 2n measurements up to a natural gauge. We also show that one can recover the full first-order term for a related equation having no gauge invariance, and that it is possible to reduce the number of measurements if the coefficients have certain symmetries. This work extends the fixed angle scattering results of Rakesh and Salo (SIAM J Math Anal 52(6):5467–5499, 2020) and (Inverse Probl 36(3):035005, 2020) to Hamiltonians with first-order perturbations, and it is based on wave equation methods and Carleman estimates.peerReviewe

Stable factorization of the Calderón problem via the Born approximation

In this article we prove the existence of the Born approximation in the context of the radial Calderón problem for Schrödinger operators. This is the inverse problem of recovering a radial potential on the unit ball from the knowledge of the Dirichlet-to-Neumann map (DtN map from now on) of the corresponding Schrödinger operator. The Born approximation naturally appears as the linear component of a factorization of the Calderón problem; we show that the non-linear part, obtaining the potential from the Born approximation, enjoys several interesting properties. First, this map is local, in the sense that knowledge of the Born approximation in a neighborhood of the boundary is equivalent to knowledge of the potential in the same neighborhood, and, second, it is Hölder stable. This shows in particular that the ill-posedness of the Calderón problem arises solely from the linear step, which consists in computing the Born approximation from the DtN map by solving a Hausdorff moment problem. Moreover, we present an effective algorithm to compute the potential from the Born approximation and show a result on reconstruction of singularities. Finally, we use the Born approximation to obtain a partial characterization of the set of DtN maps for radial potentials. The proofs of these results do not make use of Complex Geometric Optics solutions or its analogues; they are based on results on inverse spectral theory for Schrödinger operators on the half-line, in particular on the concept of A-amplitude introduced by Barry Simon