Asymptotics of the density of states of hypoelliptic almost periodic operators
Sbornik. Mathematics, Tome 34 (1978) no. 4, pp. 425-447 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For a hypoelliptic differential operator with almost periodic coefficients asymptotics are found for the density of states, complete with a remainder term. Under certain conditions on the symbol of the operator the asymptotics imply an estimate on the length of the gaps in the given operator acting in $L^2(\mathbf R^n)$. Bibliography: 25 titles.
@article{SM_1978_34_4_a1,
     author = {V. I. Bezyaev},
     title = {Asymptotics of the density of states of hypoelliptic almost periodic operators},
     journal = {Sbornik. Mathematics},
     pages = {425--447},
     year = {1978},
     volume = {34},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1978_34_4_a1/}
}
TY  - JOUR
AU  - V. I. Bezyaev
TI  - Asymptotics of the density of states of hypoelliptic almost periodic operators
JO  - Sbornik. Mathematics
PY  - 1978
SP  - 425
EP  - 447
VL  - 34
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1978_34_4_a1/
LA  - en
ID  - SM_1978_34_4_a1
ER  - 
%0 Journal Article
%A V. I. Bezyaev
%T Asymptotics of the density of states of hypoelliptic almost periodic operators
%J Sbornik. Mathematics
%D 1978
%P 425-447
%V 34
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1978_34_4_a1/
%G en
%F SM_1978_34_4_a1
V. I. Bezyaev. Asymptotics of the density of states of hypoelliptic almost periodic operators. Sbornik. Mathematics, Tome 34 (1978) no. 4, pp. 425-447. http://geodesic.mathdoc.fr/item/SM_1978_34_4_a1/

[1] V. N. Gorchakov, “Ob asimptoticheskikh svoistvakh spektralnoi funktsii gipoellipticheskikh operatorov”, DAN SSSR, 160:4 (1965), 746–749 | Zbl

[2] V. Yu. Kiselev, “Pochti-periodicheskie integralnye operatory Fure i nekotorye ikh prilozheniya”, Trudy seminara im. I. G. Petrovskogo, 1977, no. 3, 81–97 | MR

[3] B. M. Levitan, Pochti-periodicheskie funktsii, Gostekhizdat, Moskva, 1953

[4] V. N. Tulovskii, M. A. Shubin, “Ob asimptoticheskom raspredelenii sobstvennykh znachenii psevdodifferentsialnykh operatorov v $R^n$”, Matem. sb., 92(134) (1973), 571–588 | MR | Zbl

[5] V. I. Feigin, “Novye klassy psevdodifferentsialnykh operatorov v $R^n$ i nekotorye prilozheniya”, Trudy Mosk. matem. ob-va, XXXVI (1976), 594–614 | MR

[6] V. I. Feigin, “Asimptoticheskoe raspredelenie sobstvennykh chisel dlya gipoellipticheskikh sistem v $R^n$”, Matem. sb., 99(141) (1976), 594–614 | MR | Zbl

[7] V. I. Feigin, “O spektralnoi asimptotike dlya kraevykh zadach i asimptotike otritsatelnogo spektra”, DAN SSSR, 232:6 (1977), 1269–1272 | MR

[8] V. I. Feigin, “O nepreryvnom spektre differentsialnykh operatorov”, Funkts. analiz, 11:1 (1977), 43–54 | MR | Zbl

[9] L. Khërmander, “Spektralnaya funktsiya ellipticheskogo operatora”, Matematika, 13:6 (1969), 114–137

[10] M. A. Shubin, “Differentsialnye i psevdodifferentsialnye operatory v prostranstvakh pochti-periodicheskikh funktsii”, Matem. sb., 95(137):4(12) (1974), 560–587 | MR | Zbl

[11] M. A. Shubin, “Teoremy o sovpadenii spektrov psevdodifferentsialnogo pochti-periodicheskogo operatora v prostranstvakh $L^2(R^n)$ i $B^2(R^n)$”, Sib. matem. zh., 17:1 (1975), 200–215

[12] M. A. Shubin, “Ellipticheskie pochti-periodicheskie operatory i algebry fon Neimana”, Funk. analiz, 9:1 (1975), 89–90 | MR | Zbl

[13] M. A. Shubin, “Psevdodifferentsialnye pochti-periodicheskie operatory i algebry fon Neimana”, Trudy Mosk. matem. ob-va, XXXV (1976), 103–164

[14] M. A. Shubin, “Teorema Veilya dlya operatora Shredingera s pochti-periodicheskim potentsialom”, Vestnik MGU, seriya matem. i mekh., 1976, no. 2, 84–88 | Zbl

[15] M. A. Shubin, “Plotnost sostoyanii samosopryazhennykh ellipticheskikh operatorov s pochti-periodicheskimi koeffitsientami”, Trudy seminara im. I. G. Petrovskogo, 1977, no. 3, 243–275

[16] M. Burnat, A. Palczewski, “On the spectral properties of the operator $-\nabla u+q(x^1,x^2,x^3)u$ with almost periodic $q(x^1,x^2,x^3)$”, Bull. Acad. pol. Sci., Ser. Math., astron. and phis., 21:10 (1973), 917–923 | MR | Zbl

[17] L. A. Coburn, R. D. Moyer, I. M. Singer, “$C^*$-algebras of almost periodic pseudo-differential operators”, Acta Math., 130:3–4 (1973), 279–307 | DOI | MR | Zbl

[18] J. Dixmier, Les algebrès d'operateurs dans l'espace hilbertien (algebrès de von Neumann), Gauthièr-Villars, Paris, 1969 | MR

[19] H. Kumano-go, “Algebras of pseudo-differential operators”, J. Fac. Sci. Univ. Tokyo, Sec. 1A, 17:1–2 (1970), 31–50 | MR | Zbl

[20] H. Kumano-go, “Remarks on pseudo-differential operators”, J. Math. Soc. Japan, 21 (1969), 413–439 | Zbl

[21] H. Kumano-go, C. Tsutsumi, “Complex powers of hypoelliptic pseudo-differential operators with applications”, Osaka J. Math., 10 (1973), 147–174 | MR

[22] G. Mètivier, “Fonction spectrale et valeurs propres d'opérateurs non elliptiques”, C. r. Acad. Scient. Paris, 283:7 (1976), A453–A456

[23] N. Nilsson, “Asymptotic estimates for spectral functions connected with hypoelliptic differential operators”, Ark. Math., 5:6 (1965), 527–540 | DOI | MR | Zbl

[24] N. Nilsson, “Some estimates for spectral functions connected with formally hypoelliptic differential operators”, Ark. Math., 10:2 (1972), 251–275 | DOI | MR | Zbl

[25] A. Tsutsumi, “On the asymptotic behavior of resolvent kernels and spectral functions for some class of hypoelliptic operators”, J. Diff. Equat., 18 (1975), 366–385 | DOI | MR | Zbl