**Физика Низких Температур: ***Том ***43**, Выпуск **1** (Январь 2017), c. 122-131
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### The confinement of an annealed branched polymer by a potential well

**Alexander Y. Grosberg**^{1}, Joshua Kelly^{2}, and Robijn Bruinsma^{2,3}

^{1}Department of Physics and Center for Soft Matter Research, New York University 4 Washington Place, New York NY 10003, USA

E-mail: ayg1@nyu.edu

^{2}Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA

^{3}Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095, USA

*Received July 27, 2016*

**Аннотация**

The Lifshitz equation for the confinement of a linear polymer in a spherical cavity of radius R has the form of the Schrödinger equation for a quantum particle trapped in a potential well with flat bottom and infinite walls at radius *R*. We show that the Lifshitz equation of a confined annealed branched polymer has the form of the Schrödinger equation for a quantum harmonic oscillator. The harmonic oscillator potential results from the repulsion of the many branches from the potential walls. Mathematically, it must be obtained from the solution of the equation of motion of a second, now classical, particle in a non-linear potential that depends self-consistently on the eigenvalue of the quantum oscillator. The resulting confinement energy has a 1/*R*^{4} dependence on the confinement radius *R*, in agreement with scaling arguments. We discuss the application of this result to the problem of the confinement of single-stranded RNA molecules inside spherical capsids.

PACS: 36.20.–r Macromolecules and polymer molecules;

PACS: 87.15.H– Dynamics of biomolecules.

**Ключевые слова: **confined branched polymer, Lifshitz equation.

**Published online:** *November 25, 2016*