### A boundary value problem for the wave equation.

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We study a two-particle quantum system given by a test particle interacting in three dimensions with a harmonic oscillator through a zero-range potential. We give a rigorous meaning to the Schrödinger operator associated with the system by applying the theory of quadratic forms and defining suitable families of self-adjoint operators. Finally we fully characterize the spectral properties of such operators.

A non-homogeneous Hardy-like inequality has recently been found to be closely related to the knowledge of the lowest eigenvalue of a large class of Dirac operators in the gap of their continuous spectrum.

For simply connected planar domains with the maximal conformal radius 1 it was proven in 1954 by G. Pólya and M. Schiffer that for the eigenvalues λ of the fixed membrane for any n the following inequality holds [...] where λ(o) are the eigenvalues of the unit disk. The aim of the paper is to give a sharper version of this inequality and for the sum of all reciprocals to derive formulas which allow in some cases to calculate exactly this sum.

We study the mathematical properties of a general model of cell division structured with several internal variables. We begin with a simpler and specific model with two variables, we solve the eigenvalue problem with strong or weak assumptions, and deduce from it the long-time convergence. The main difficulty comes from natural degeneracy of birth terms that we overcome with a regularization technique. We then extend the results to the case with several parameters and recall the link between this...