The wave functions and the Ps energy of the center of gravity motion, respectively, in the 2D case can then be obtained: (40) (41) Next, consider the relative motion of the electron-positron pair. Seeking the wave functions of the problem in the form , after some transformations, the radial part of the reduced Schrodinger PARP inhibitor equation can be written as: (42) At ξ Selleckchem C646 → 0, the solution of (42) sought in the form χ(ξ → 0) = χ 0 ~ ξ λ [45, 46]. Here, in contrast to Equation 21, the quadratic equation is obtained with the following solutions: (43) In the 2D case, the solution satisfying the condition of finiteness of

the wave function is given as . At ξ → ∞, proceeding analogously to the solution of Equation 21, one should again arrive find more at the equation of Kummer (24) but with different parameter λ. Finally, for the energy of the 2D Ps with Kane’s dispersion law one can get: (44) A similar result for the case of a parabolic dispersion law is written as: (45) Here N ′  = n r + |m| is Coulomb

principal quantum number for Ps. Again, determining the binding energy as the energy difference between cases of presence and absence of positron in a QD, one finally obtains the expression: (46) In the case of free 2D Ps with Kane’s dispersion law, the energy is: (47) Here again, the expression (47) follows from (44) at the limit r 0 → ∞. Define again the confinement energy in the 2D case as the difference between the absolute values of the Ps energy in a circular QD and a free Ps energy: (48) Here, it is also necessary to note two remarks. First, in contrast to the 3D Ps case, all states with m = 0 are unstable in a semiconductor with Kane’s dispersion law. It is also important that instability is the consequence not only of the dimension reduction of the sample but also of the change of the dispersion law. In other words, ‘the particle falling into center’ [45] or, more correctly, the annihilation Thymidine kinase of the

pair in the states with m = 0 is the consequence of interaction of energy bands. Thus, the dimension reduction leads to the fourfold increase in the Ps ground-state energy in the case of parabolic dispersion law, but in the case of Kane’s dispersion law, annihilation is also possible. Note also that the presence of SQ does not affect the occurrence of instability as it exists both in the presence and in the absence of SQ (see (44) and (47)). Second, the account of the bands’ interaction removes the degeneracy of the magnetic quantum number. However, the twofold degeneracy of m of energy remains. Thus, in the case of Kane’s dispersion law, the Ps energy depends on m 2, whereas in the parabolic case, it depends on |m|. Due to the circular symmetry of the problem, the twofold degeneracy of energy remains in both cases of dispersion law. Results and discussion Let us proceed to the discussion of results.