×

On a trace formula of the Buslaev-Faddeev type for a long-range potential. (English) Zbl 0952.34069

The author sketches a procedure for deriving a trace formula for \(-d^2/dx^2+v(x)\) on \((0, +\infty[\), with Dirichlet condition at \(0\), where \(|v^{(n)}(x)|\leq Q (1+x)^{-\alpha-n}\) with \(\alpha>1/2\) and \(n=0\), 1, 2, 3, 4. More general results and counter-examples showing the role of the various assumptions are announced. The results rely heavily on the paper by L. S. Koplienko [Sib. Math. J. 26, 365-369 (1985; Zbl 0623.47060)].

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L25 Scattering theory, inverse scattering involving ordinary differential operators
47A55 Perturbation theory of linear operators
47E05 General theory of ordinary differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Citations:

Zbl 0623.47060
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Gel’fand I. M., Dokl. Akad. Nauk SSSR 88 pp 593– (1953)
[2] DOI: 10.1090/trans2/018/06 · Zbl 0098.28701
[3] Buslaev B. S., Dokl. Akad. Nauk SSSR 132 pp 13– (1960)
[4] Buslaev B. S., Sov. Math. Dokl. 1 pp 451– (1960)
[5] Birman M. Sh., St. Petersburg Math. J. 4 pp 833– (1993)
[6] DOI: 10.1070/SM1993v074n02ABEH003360 · Zbl 0774.34062
[7] DOI: 10.1007/BF02547335 · Zbl 0885.34070
[8] DOI: 10.1007/BF02099586 · Zbl 0821.34076
[9] DOI: 10.1142/S0129055X95000347 · Zbl 0833.34084
[10] Rybkin A. V., Russ. Acad. Sci. Sb. Math. 83 pp 237– (1995)
[11] DOI: 10.1215/S0012-7094-96-08322-2 · Zbl 0876.47010
[12] DOI: 10.1007/BF01028668
[13] DOI: 10.1007/BF01028668
[14] DOI: 10.1007/BF01035543 · Zbl 0625.35021
[15] DOI: 10.1007/BF01035543 · Zbl 0625.35021
[16] DOI: 10.1007/BF01017012
[17] DOI: 10.1007/BF01017012
[18] DOI: 10.1007/BF02790222 · Zbl 0807.35169
[19] DOI: 10.1007/BF00968623 · Zbl 0623.47060
[20] DOI: 10.1007/BF00968623 · Zbl 0623.47060
[21] DOI: 10.1007/BF01090727
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.