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Laplace and Schr\"odinger operators on regular metric trees: the discrete spectrum case

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arxiv math/0111023 v1 pith:HI2OS35M submitted 2001-11-02 math.SP math-phmath.MP

Michael Solomyak This is my paper · ORCID
classification math.SPmath-phmath.MP
keywords odingeroperatorsschrtreedecompositionmetricoperatorregular
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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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  1. Extinction and propagation phenomena for semilinear parabolic equations on metric trees

    math.AP 2025-05 conditional novelty 6.0 of 10

    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).

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