Geodesic
The Spectrum of Resonances and the Trace Formula in a Potential Scattering Problem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 79-89.

Voir la notice de l'article provenant de la source Math-Net.Ru

For the wave equation with a potential perturbation, the localization and the asymptotic distribution of resonances, i.e., poles of the scattering matrix, are studied. To this end, it proves useful to exploit the relation between these poles and the spectrum of the corresponding Lax–Phillips semigroup. An explicit description of the generator of this semigroup is given. The regularized trace formula for the operators specifying the evolution of the initial data is applied to estimate how rarefied the spectrum of resonances can be.
Keywords: resonance, scattering matrix, Lax–Phillips semigroup
Mots-clés : trace formula. 
@article{FAA_2004_38_3_a6,
     author = {S. A. Stepin},
     title = {The {Spectrum} of {Resonances} and the {Trace} {Formula} in a {Potential} {Scattering} {Problem}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {79--89},
     publisher = {mathdoc},
     volume = {38},
     number = {3},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a6/}
}
TY  - JOUR
AU  - S. A. Stepin
TI  - The Spectrum of Resonances and the Trace Formula in a Potential Scattering Problem
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2004
SP  - 79
EP  - 89
VL  - 38
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a6/
LA  - ru
ID  - FAA_2004_38_3_a6
ER  - 
%0 Journal Article
%A S. A. Stepin
%T The Spectrum of Resonances and the Trace Formula in a Potential Scattering Problem
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2004
%P 79-89
%V 38
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a6/
%G ru
%F FAA_2004_38_3_a6
S. A. Stepin. The Spectrum of Resonances and the Trace Formula in a Potential Scattering Problem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 79-89. http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a6/

[1] Laks P., Fillips R., Teoriya rasseyaniya, Mir, M., 1972

[2] Vainberg B. R., “O sobstvennykh funktsiyakh operatora, otvechayuschikh polyusam analiticheskogo prodolzheniya rezolventy cherez nepreryvnyi spektr”, Matem. sb., 87:2 (1972), 293–308 | MR | Zbl

[3] Melrose R., Geometric scattering theory, Cambridge University Press, 1995 | MR | Zbl

[4] Lidskii V. B., “Nesamosopryazhennye operatory, imeyuschie sled”, DAN SSSR, 125:3 (1959), 485–488 | MR

[5] Arsenev A. A., Singulyarnye potentsialy i rezonansy, Izd-vo MGU, M., 1974 | MR

[6] Baz A. I., Zeldovich Ya. B., Perelomov A. M., Rasseyanie, reaktsii i raspady v nerelyativistskoi kvantovoi mekhanike, Nauka, M., 1971 | Zbl

[7] Buslaev V. S., “Formuly sledov dlya operatora Shrëdingera v trekhmernom prostranstve”, DAN SSSR, 143:5 (1962), 1067–1070 | MR | Zbl

[8] Pavlov B. S., “Spektralnyi analiz dissipativnogo operatora Shrëdingera v terminakh funktsionalnoi modeli”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 65, VINITI, M., 1991, 95–163

[9] Sjöstrand J., “A trace formula and review of some estimates for resonances”, Microlocal analysis and spectral theory, Math. Phys. Sci., 490, Kluwer Acad. Publ., 1997, 377–437 | MR | Zbl

[10] Zworski M., “Quantum resonances and partial differential equations”, Proc. ICM, vol. 3, 2002, 243–252 | MR | Zbl

[11] de Alfaro V., Redzhe T., Potentsialnoe rasseyanie, Mir, M., 1966 | Zbl

[12] Stepin S. A., “Volnovye operatory dlya linearizovannogo uravneniya Boltsmana v odnoskorostnoi teorii perenosa”, Matem. sb., 192:1 (2001), 139–160 | DOI | MR | Zbl

[13] Bardos C., Guillot J. C., Ralston J., “La relation de Poisson pour l'equation des ondes dans un ouvert non borne. Application a la theorie de la diffusion”, Comm. Partial Differential Equations, 7:8 (1982), 905–958 | DOI | MR | Zbl

[14] Khërmander L., Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, t. 3, Mir, M., 1987 | MR

[15] Babich V. M., “Metod Adamara i asimptotika spektralnoi funktsii differentsialnogo operatora vtorogo poryadka”, Matem. zametki, 28:5 (1980), 689–694 | MR | Zbl

[16] Melrose R., “Scattering theory and the trace of the wave group”, J. Funct. Anal., 45 (1982), 29–40 | DOI | MR | Zbl

[17] Sá Barreto A., Zworski M., “Existence of resonances in potential scattering”, Comm. Pure Appl. Math., 49 (1996), 1271–1280 | 3.0.CO;2-7 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl