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PACS 25.60.Pj -Fusion reactions PACS 25.30.Mr -Muon-induced reactions (including the EMC effect) PACS 36.10.-k -Exotic atoms and molecules (containing mesons, antiprotons and other unusual particles)Abstract -The basic principle of muon catalyzed fusion is that it is possible to obtain energy with temperatures much lower than that required for thermal nuclear fusion. The ground-state energies of muonic molecules formed by ionized proton-proton, deuterium-deuterium and tritium-tritium nuclei plus a negative muon confined in a two-dimensional spatial region are investigated, assuming a ln(r) electrostatic potential instead of the Coulomb potential 1/r frequently used. The effective two-dimensional potential of these molecules is analytically calculated within a quasi-adiabatic approximation, and the probability of fusion is numerically computed for some molecules. In this letter some unexpected theoretical results are given and compared to those of the same molecules described in three dimensions, using the same approach. For example, for the ddμ molecule, the fusion rate is of the order of 10 8 times greater than the predicted value in 3D. For the same molecule, the tunneling rate is also amplified by a factor 10 4 .

In this paper, a quantum dot mathematical model based on a twodimensional Schrödinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun equation have some limitations. The known polynomial solutions are confronted with new numerical calculations based on the Numerov method. A good qualitative agreement between them emerges. The numerical method being more general gives rise to new solutions. In particular, we are now able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound state for such planar system, in the case = 0, is predicted and its respective eigenvalue is determined. Keywords Quantum dot model · Numerov numerical method · two-electron system · Schrödinger equation PACS PACS 81.07.Ta · 78.67.Hc · 36.10.-k 1 IntroductionModern technics in nanometer-scale semiconductor manufacturing enable the creation of quantum confinement of only a few electrons. These few-body systems are often called quantum dots [1]. They can be described by a model F. Caruso