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Spectral analysis of perturbed multiplication operators occurring in polymerization chemistry. (English) Zbl 0714.45013

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1989, Exp. No. 19, 6 p. (1989).
In the kinetic model for polymerization there appears the boundary value problem: \[ \frac{\partial P}{\partial t}(x,t)=- (x+2)P(x,t)+2\int^{\infty}_{x}x/yP(y,t)dy+2\int^{x}_{0}e^{y- x}P(y,t)dy,x,t>0; \]
\[ P(.,0)=P_ 0\in \{=f\geq 0| \quad \int^{\infty}_{0}f(y)dy=1\}, \] where P(x,t) is the probability distribution at time t for the length of the polymer molecule. In the study of this boundary value problem, the author himself obtained some results in previous papers, namely existence and uniqueness of the solution, and asymptotic behaviour of the solution.
In the present paper he establishes the selfadjoint operator corresponding to the above equation and studies its spectrum. The principal result: This selfadjoint operator has purely absolutely continuous spectrum \([0,\infty)\) and infinitely many negative eigenvalues converging to zero.
Reviewer: St.Zanfir

MSC:

45K05 Integro-partial differential equations
45C05 Eigenvalue problems for integral equations
45M05 Asymptotics of solutions to integral equations
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