Levinson’s theorem for graphs. II. (English) Zbl 1278.81093
Summary: [For part I see the authors, ibid. 52, No. 8, 082102, 9 p. (2011; Zbl 1272.81081)] We prove Levinson’s theorem for scattering on an \((m+n)\)-vertex graph with n semi-infinite paths each attached to a different vertex, generalizing a previous result for the case \(n = 1\). This theorem counts the number of bound states in terms of the winding of the determinant of the S-matrix. We also provide a proof that the bound states and incoming scattering states of the Hamiltonian together form a complete basis for the Hilbert space, generalizing another result for the case \(n = 1\).{
©2012 American Institute of Physics}
©2012 American Institute of Physics}
MSC:
81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
81U05 | \(2\)-body potential quantum scattering theory |
81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
81U20 | \(S\)-matrix theory, etc. in quantum theory |
34L25 | Scattering theory, inverse scattering involving ordinary differential operators |
Citations:
Zbl 1272.81081References:
[1] | DOI: 10.1063/1.1666364 · doi:10.1063/1.1666364 |
[2] | DOI: 10.1103/PhysRevLett.102.180501 · doi:10.1103/PhysRevLett.102.180501 |
[3] | DOI: 10.1145/780542.780552 · Zbl 1192.81059 · doi:10.1145/780542.780552 |
[4] | DOI: 10.1063/1.3622608 · Zbl 1272.81081 · doi:10.1063/1.3622608 |
[5] | DOI: 10.4086/toc.2008.v004a008 · Zbl 1213.68284 · doi:10.4086/toc.2008.v004a008 |
[6] | DOI: 10.1103/PhysRevA.58.915 · doi:10.1103/PhysRevA.58.915 |
[7] | DOI: 10.1137/0522045 · Zbl 0822.47032 · doi:10.1137/0522045 |
[8] | Kato T., Perturbation Theory for Linear Operators (1966) · Zbl 0148.12601 |
[9] | DOI: 10.1103/PhysRev.75.1445 · doi:10.1103/PhysRev.75.1445 |
[10] | DOI: 10.1103/PhysRevA.80.052330 · doi:10.1103/PhysRevA.80.052330 |
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