Inverz szóráselméleti kutatások = Problems of inverse scattering theory

Kifejlesztettünk egy új fix-energiás inverz kvantum szórás módszert. Szórási adatok invertálására alkalmassá tettük a Cox-Thompson inverz kvantum szórás eljárást. Bose-kondenzátumok ütközéséből származó fázistolás adatokból határoztunk meg effektív Rb-Rb atomi potenciálokat. Kétkomponensű Bose-Einstein kondenzátumok stabilitását vizsgáltuk. Megbecsültük a különző specimenek közti szóráshosszak azon tartományát, amely esetében szoliton gerjesztések létrejöhetnek a kondenzátumban. Kifejelesztettünk és numerikusan teszteltünk egy csatolt Gross-Pitaevskii egyenlet megoldó programot. A kutatási munkatervben vállalt 7 publikáció és 4 konferencia előadást jelentősen túl teljestettük, amennyiben 3 disszertáció, 3 konferenciakiadvány és 25 publiáció született az egy évvel meghosszabbított, 5 éves 4 résztvevős kutatás alatt. Ezen kívül egy nemzetközi inverz kvantum szórás konferenciát is rendeztünk (www.math.bme.hu/~hirvath/iqs). | New fix-energy inverse quantum scattering method has been developed. The Cox-Thompson inverse quantum scattering procedure has been made appropriate to invert scattering data. We have determined Rb-Rb atomic scattering potential from data extracted from Bose condensate collisions. Stability of two-component Bose-Einstein condensates has been inversigated. Assessments have been given to values of interspecies scattering length at which soliton excitations are to be expected to exist inside the condensate. We have developed and numerically tested an evolution code which simulate the time evolution of a two-component Bose-Einstein condensate accoring to the Gross-Pitaevskii equation. The original undertaking has been well overcompleted in that 3 theses (1 Phd and 2 DSc), 3 conference contribution and 25 publications in journals of high international reputation have been delivered ba the 4 participants during the 5 years research. Besides also an international inverse quantum scattering conference has been held (www.math.bme.hu/~hirvath/iqs)

Quantum chaos in one dimension?

In this work we investigate the inverse of the celebrated Bohigas-Giannoni-Schmit conjecture. Using two inversion methods we compute a one-dimensional potential whose lowest N eigenvalues obey random matrix statistics. Our numerical results indicate that in the asymptotic limit, N->infinity, the solution is nowhere differentiable and most probably nowhere continuous. Thus such a counterexample does not exist.Comment: 7 pages, 10 figures, minor correction, references extende

Painlev\'{e} test of coupled Gross-Pitaevskii equations

Painlev\'{e} test of the coupled Gross-Pitaevskii equations has been carried out with the result that the coupled equations pass the P-test only if a special relation containing system parameters (masses, scattering lengths) is satisfied. Computer algebra is applied to evaluate j=4 compatibility condition for admissible external potentials. Appearance of an arbitrary real potential embedded in the external potentials is shown to be the consequence of the coupling. Connection with recent experiments related to stability of two-component Bose-Einstein condensates of Rb atoms is discussed.Comment: 13 pages, no figure

Quantum mechanical potentials related to the prime numbers and Riemann zeros

Prime numbers are the building blocks of our arithmetic, however, their distribution still poses fundamental questions. Bernhard Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the non-trivial zeros of the Riemann $\zeta(s)$ function. According to the Hilbert-P{\'o}lya conjecture there exists a Hermitean operator of which the eigenvalues coincide with the real part of the non-trivial zeros of $\zeta(s)$. This idea encourages physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Mar{\v{c}}henko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the $\zeta(s)$ function. We demonstrate the multifractal nature of these potentials by measuring the R{\'e}nyi dimension of their graphs. Our results offer hope for further analytical progress.Comment: 7 pages, 5 figures, 2 table

Physics of the Riemann Hypothesis

Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics and also how physics may be suggestive for the resolution of one of mathematics' most famous unconfirmed conjectures, the Riemann Hypothesis. Does physics hold an essential key to the solution for this more than hundred-year-old problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its pole-structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader.Comment: 27 pages, 9 figure

Conceptual coherence of non-Newtonian worldviews in Force Concept Inventory data

The Force Concept Inventory is one of the most popular and most analyzed multiple-choice concept tests used to investigate students’ understanding of Newtonian mechanics. The correct answers poll a set of underlying Newtonian concepts and the coherence of these underlying concepts has been found in the data. However, this inventory was constructed after several years of research into the common preconceptions held by students and using these preconceptions as distractors in the questions. Their sole purpose is to deflect non-Newtonian candidates away from the correct answer. Alternatively, one can argue that the responses could also be treated as polling these preconceptions. In this paper we shift the emphasis of the analysis away from the correlation structure of the correct answers and look at the latent traits underlying the incorrect responses. Our analysis models the data employing exploratory factor analysis, which uses regularities in the data to suggest the existence of underlying structures in the cognitive processing of the students. This analysis allows us to determine whether the data support the claim that there are alternate non-Newtonian worldviews on which students’ incorrect responses are based. The existence of such worldviews, and their coherence, could explain the resilience of non-Newtonian preconceptions and would have significant implications to the design of instruction methods. We find that there are indeed coherent alternate conceptions of the world which can be categorized using the results of the research that led to the construction of the Force Concept Inventory

Central distractors in Force Concept Inventory data

The Force Concept Inventory was designed to poll the Newtonian conception of force. While there are many in-depth studies analyzing response data that look at the structure of the correct answers, we believe that the incorrect answers also carry revealing information about the students’ worldview. The inventory was originally designed so that the “distractors” in each question reflected commonly held misconceptions, and thus the rate at which students guess the correct answer is very low. Students select incorrect answers that correspond to the misconception that they hold and there are very few responses which appear obviously wrong to students. A side effect of this approach is that the incorrect responses reflect the non-Newtonian worldviews held by students. These non-Newtonian worldviews are coherent and robust, and this, at least in part, helps to explain why these misconceptions are so resistant to instruction. In this study we focus once more on the misconception data in Force Concept Inventory responses, particularly on the linkages between these misconceptions. We hypothesize that there are distinct groupings of distractor items formed by the strength of the association between these items. The two largest groupings are associated with the “impetus” world view in which the motion of an object is determined by the quantity of impetus which that object contains. We find that certain central items hold particularly important places in these groupings and also that individual groupings may be connected to each other by “connector” items. We finally suggest that, on the basis of this study, that these non-Newtonian worldviews might best be dismantled by addressing these key central and connector items