On periodic continuants

MHD α 2 -dynamo, Squire equation and PT -symmetric interpolation between square well and harmonic oscillatorWe propose two ways for determining the Green's matrix for problems admitting Hamiltonians that have infinite symmetric tridiagonal ͑i.e., Jacobi͒ matrix form on some basis representation. In addition to the recurrence relation coming from the Jacobi-matrix, the first approach also requires the matrix elements of the Green's operator between the first elements of the basis. In the second approach the recurrence relation is solved directly by continued fractions and the solution is continued analytically to the whole complex plane. Both approaches are illustrated with the non-trivial but calculable example of the D-dimensional Coulomb Green's matrix. We give the corresponding formulas for the D-dimensional harmonic oscillator as well.

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“…From the theory of special matrices we know that the inverse of a Jacobi-matrix, i.e. the Green's matrix, possesses the property [15]…”

Section: )mentioning

confidence: 99%

Green’s matrix from Jacobi-matrix Hamiltonian

Kónya

Lévai

Papp

1997

Journal of Mathematical Physics

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“…Formula can even be extended to some generalizations of Toeplitz matrices, as we can find in other works …”

Section: K‐tridiagonal Matricesmentioning

confidence: 91%

“…Therefore, these three matrices share exactly the same spectra. Furthermore, this symmetrization process is not restricted to tridiagonal Toeplitz matrices as we can find in the work of Rózsa . In fact, it can be extended to any acyclic matrix, that is, a matrix whose graph is cycle free (cf.…”

Section: Symmetrizationmentioning

confidence: 99%

“…Formula (3) can even be extended to some generalizations of Toeplitz matrices, as we can find in other works. [6][7][8] We can also provide an interpretation of (3) in terms of graph theory. In fact, if we seeÃ n as the (weighed) adjacency matrix of a graph G, such graph consists of two types of (disconnect) paths.…”

Section: K-tridiagonal Matricesmentioning

confidence: 99%

“…Furthermore, this symmetrization process is not restricted to tridiagonal Toeplitz matrices as we can find in the work of Rózsa. 8 In fact, it can be extended to any acyclic matrix, that is, a matrix whose graph is cycle free (cf. other works 15,16 ) That is, any acyclic matrix can be symmetrized keeping the same spectrum.…”

Section: Symmetrizationmentioning

confidence: 99%

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