Some general properties of the wave functions of complex-valued potentials with real spectrum are studied. The main results are presented in a series of lemmas, corollaries and theorems that are satisfied by the zeros of the real and imaginary parts of the wave functions on the real line. In particular, it is shown that such zeros interlace so that the corresponding probability densities ρ(x) are never null. We find that the profile of the imaginary part V I (x) of a given complex-valued potential determines the number and distribution of the maxima and minima of the related probability densities. Our conjecture is that V I (x) must be continuous in R, and that its integral over all the real line must be equal to zero in order to get control on the distribution of the maxima and minima of ρ(x). The applicability of these results is shown by solving the eigenvalue equation of different complex potentials, these last being either PT -symmetric or not invariant under the PT -transformation.

No abstract

The lens is a complex optical component of the human eye because of its physiological structure: the surface is aspherical and the structural entities create a gradient refractive index (GRIN). Most existent models of the lens deal with its external shape independently of the refractive index and, subsequently, through optimization processes, adjust the imaging properties. In this paper, we propose a physiologically realistic GRIN model of the lens based on a single function for the whole lens that accurately describes different accommodative states simultaneously providing the corresponding refractive index distribution and the external shape of the lens by changing a single parameter that we associate with the function of the ciliary body. This simple, but highly accurate model, is incorporated into a schematic eye constructed with reported experimental biometric data and accommodation is simulated over a range of 0 to 6 diopters to select the optimum levels of image quality. Changes with accommodation in equatorial and total axial lens thicknesses, as well as aberrations, are found to lie within reported biometric data ranges.

The COVID-19 confinement has represented both opportunities and losses for education. Rarely before has any other period moved the human spirit into such discipline or submission—depending on one’s personal and emotional points of view. Both extremes have been widely influenced by external factors on each individual’s life path. Education in the sciences and engineering has encountered more issues than other disciplines due to specialized mathematical handwriting, experimental demonstrations, abstract complexity, and lab practices. This work analyses three aspects of science education courses taught by university professors in a collaborative teacher cluster, sharing technology applications and education methodologies in science over three semesters when the COVID-19 lockdown was in effect. The first aspect was a didactic design coming from several educational frameworks through adoption or sharing. The second one was an analysis by discipline of multiple factors affecting student engagement during the health contingency. The third analysis examined the gains and losses in our students caused by the university closure and the pandemic’s intrusions. The report explores the correlations of the exiting student perceptions with their academic performance in the courses and survey results about the impact of decisions or happenings during the crisis. This work’s value lies in the lessons for the future of education concerning the teacher dominions of didactic design, support, and collaboration in a broader sense than only for teaching.

To the best of our knowledge, at the present time there is no answer to the fundamental question stated in the title that provides a complete and satisfactory physical description of the structured nature of Hermite-Gauss beams. The purpose of this manuscript is to provide proper answers supported by a rigorous mathematical-physics framework that is physically consistent with the observed propagation of these beams under different circumstances. In the process we identify that the paraxial approximation introduces spurious effects in the solutions that are unphysical. By removing them and using the property of self-healing, that is characteristic to structured beams, we demonstrate that Hermite-Gaussian beams are constituted by the superposition of four traveling waves.