On the spectral theory of second-order elliptic differential operators

Yu. A. Semenov

Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 221-247

Résumé

$(L^p,\,L^q)$-estimates and $\mathfrak S_p$-properties of powers of the resolvent in weighted $L^p$-spaces are studied for a certain selfadjoint realization of the formal differential expression $-\sum_{k,j}\partial_k a_{kj}\partial_j+V$. A theory of generalized eigenfunction expansions is developed, and the continuous spectrum is investigated. Bibliography: 30 titles.

Bibliographie

[1] Semenov Yu. A., “Gladkost obobschennykh reshenii uravneniya $Hu=f$ i suschestvennaya samosopryazhennost operatora $H=-\nabla_ia_{ij}+V$ s izmerimymi koeffitsientami”, Matem. sb., 127(169) (1985), 311–335 | MR

[2] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[3] Kato T., “Remarks on the selfadjointness and related problems to differential operators”, Spectral Theory of Differential operators, North-Holland Publishing Company, 1981, 253–256 | MR

[4] Kovalenko V. F., Semenov Yu. A., “Nekotorye voprosy razlozheniya po obobschennym, sobstvennym funktsiyam operatora Shredingera s silno singulyarnymi potentsialami”, UMN, 33:4 (1978), 107–140 | MR | Zbl

[5] Kovalenko V. F., Perelmuter M. A., Semenov Yu. A., “Schrödinger operators with $L_{w}{}^{l/2}(\mathbf{R}^l)$-potentials”, J. Math. Phys., 22 (1981), 1033–1044 | DOI | MR | Zbl

[6] Herbst I., Sloan A., “Perturbations of translation invariant positivity preserving semigroups in $L^2(\mathbf{R}^N)$”, Trans. Amer. Math. Soc., 236 (1978), 325–360 | DOI | MR | Zbl

[7] Bukhvalov A. V., “Ob integralnom predstavlenii lineinykh operatorov”, Zap. nauchn. seminara LOMI, 47 (1974), 5–14

[8] Korotkov V. B., Integralnye operatory, Izd-vo Novosibirskogo un-ta, Novosibirsk, 1977 | MR

[9] Bukhvalov A. V., “Prilozheniya metodov teorii poryadkovo ogranichennykh operatorov k teorii operatorov v prostranstvakh $L^p$”, UMN, 38:6 (1983), 37–83 | MR | Zbl

[10] Iosida K., Funktsionalnyi analiz, Mir, M., 1967 | MR

[11] Schechter M., “Scattering theory for the Schrödinger equation with potentials not of short range”, Vekua yubilee volume, 1977

[12] Aizenman M., Simon B., “Brownian motion and Harnack inequality for Schrödinger operators”, Comm. Pure and Appl. Math., 35:2 (1982), 209–273 | DOI | MR | Zbl

[13] Semenov Yu. A., “Gladkost obobschennykh reshenii uravneniya $\bigl(\lambda-\sum_{ij}\nabla_ia_{ij}\nabla_j\bigr)u=f$ s nepreryvnymi koeffitsientami”, Matem. sb., 118(160) (1982), 399–410 | Zbl

[14] Simon B., “Schrödinger semigroups”, Bull. A. M. S. (New series), 7:3 (1982), 447–526 | DOI | MR | Zbl

[15] Moser J., “A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations”, Comm. Pure and Appl. Math., 61 (1960), 151–170 | MR

[16] Trudinger N., “Linear elliptic operators with measurable coefficients”, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 265–308 | MR | Zbl

[17] Berezanskii Yu. M., Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Naukova dumka, Kiev, 1965 | MR

[18] Simon B., “Spectrum and continuum eigenfunctions of Schrödinger operators”, J. Func. Anal., 42 (1981), 66–83 | DOI | MR

[19] Shnol E. E., “O povedenii sobstvennykh funktsii uravneniya Shrediigera”, Matem. sb., 42(84) (1967), 273–286

[20] Simon B., “Lower semicontinuity of positive quadratic forms”, Proc. Roy. Soc. Edinburgh A, 79:3–4 (1978), 267–273 | MR

[21] Perelmuter M. A., Semenov Yu. A., “On decoupling of finite singularities in the scattering theory for the Schrödinger operators with a magnetic field”, J. Math. Phys., 22:3 (1981), 521–533 | DOI | MR | Zbl

[22] Davies E. B., “On the Enss'approach to scattering theory”, Duke Math. J., 47:1 (1980), 171–186 | DOI | MR

[23] Perelmuter M. A., “K teorii rasseyaniya dlya ellipticheskikh operatorov vtorogo poryadka”, Ukr. matem. zhurn., 34:5 (1982), 572–580 | MR

[24] Davies E. B., “Scattering from infinite sheets”, Math. Proc. Camb. Phil. Soc., 82:2 (1977), 327–334 | DOI | MR | Zbl

[25] Amrein W. O., Pearson D. B., Wollenberg M., “Evanescence of states and asymptotic completeness”, Helv. Phys. Acta, 53:3 (1980), 335–351 | MR

[26] Mourre E., “Line between the geometrical and the spectral transformations approaches in scattering theory”, Comm. Math. Phys., 68 (1979), 91–94 | DOI | MR | Zbl

[27] Simon B., “Phase space analysis of simple scattering systems: extensions of some work of Enss”, Duke Math. J., 46 (1979), 119–168 | DOI | MR | Zbl

[28] Dodds P. G., Fremlin D. H., “Compact operators in Banach lattices”, Isr. J. Math., 34 (1979), 287–320 | DOI | MR | Zbl

[29] Stein E. M., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[30] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965