SCHRÖDINGER OPERATORS ON HOMOGENEOUS METRIC TREES: SPECTRUM IN GAPS

By A. Sobolev, Michael Solomyak. 2001

75 citations recorded by the source corpus.

The paper studies the spectral properties of the Schrodinger operator AgV = A0 + gV on a homogeneous rooted metric tree, with a decaying real-valued potential V and a coupling constant g ≥ 0. The spectrum of the free Laplacian A0 = -Δ has a band-gap structure with a single eigenvalue of infinite multiplicity in the middle of each finite gap. The perturbation gV gives rise to extra eigenvalues in the gaps. These eigenvalues are monotone functions of g if the potential V has a fixed sign. Assuming that the latter condition is satisfied and that V is symmetric, i.e. depends on the distance to the root of the tree, we carry out a detailed asymptotic analysis of the counting function of the discrete eigenvalues in the limit g → ∞. Depending on the sign and decay of V, this asymptotics is either of the Weyl type or is completely determined by the behaviour of V at infinity.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/598f746ad47011f77bb976f2d8a30af086196c94

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In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Quantum graphs: I. Some basic structures (2004, background): “[3, 5, 6, 7, 8], [16]-[20], [29]-[32], [39], [43]-[49], [54]-[61], [64, 65, 74, 79, 80 ], other”
Graph Laplacians and topology (2008, background): “[27]) shows the difficulties in solving the inverse problem.”
On the spectrum of the Laplacian on regular metric trees (2004, methodology): “Now we present an example (borrowed from [13]) of a tree for which the Laplacian is positive definite.”