On regular metric trees, KPP solutions propagate if f'(0) > E0 and become extinct if f'(0) < E0 (homogeneous case), with speed estimates between 2*sqrt(f'(0)-E0) and M*rho1*b1/(b1-1).
Laplace and Schr\"odinger operators on regular metric trees: the discrete spectrum case
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abstract
The Schr\"odinger operator on a metric tree is a family of ordinary differential operators on its edges complemented by certain matching conditions at the vertices. The regular trees are highly symmetric. This allows one to construct an orthogonal decomposition of the space L_2 on the tree which reduces the Schr\"odinger operator with any symmetric weight. Using this decomposition, we analyse the spectrum of such operators, including the free Laplacian, under various assumptions about the tree and the potential.
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Extinction and propagation phenomena for semilinear parabolic equations on metric trees
On regular metric trees, KPP solutions propagate if f'(0) > E0 and become extinct if f'(0) < E0 (homogeneous case), with speed estimates between 2*sqrt(f'(0)-E0) and M*rho1*b1/(b1-1).