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Register a new userTwo solvable systems of coagulation equations with limited aggregations
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Authors
Jean Bertoin
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We consider two simple models for the formation of polymers where at the initial time, each monomer has a certain number of potential links (called arms in the text) that are consumed when aggregations occur. Loosely speaking, this imposes restrictions on the number of aggregations. The dynamics of concentrations are governed by modifications of Smoluchowskis coagulation equations. Applying classical techniques based on generating functions, resolution of quasi-linear PDEs, and Lagrange inversion formula, we obtain explicit solutions to these non-linear systems of ODEs. We also discuss the asymptotic behavior of the solutions and point at some connexions with certain known solutions to Smoluchowskis coagulation equations with additive or multiplicative kernels.
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A novel family of exactly solvable quantum systems on curved space is presented. The family is the quantum version of the classical Perlick family, which comprises all maximally superintegrable 3-dimensional Hamiltonian systems with spherical symmetry. The high number of symmetries (both geometrical and dynamical) exhibited by the classical systems has a counterpart in the accidental degeneracy in the spectrum of the quantum systems. This family of quantum problem is completely solved with the techniques of the SUSYQM (supersymmetric quantum mechanics). We also analyze in detail the ordering problem arising in the quantization of the kinetic term of the classical Hamiltonian, stressing the link existing between two physically meaningful quantizations: the geometrical quantization and the position dependent mass quantization.
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