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# Spectral Theory

MAIN GROUPS OF RESULTS

1. Spectral asymptotics for Schrödinger operators with polynomial  magnetic and electric fields

M.Boyarchenko, S. Levendorskiĭ,Generalizations of the classical Weyl and Colin de Verdiere's formulas and the orbit method  (with Mitya Boyarchenko), Proc. Nat. Acad. of Sci USA 102  (2005) , 5663-5668  pdf file

M.Boyarchenko, S. Levendorskiĭ,Beyond the classical Weyl and Colin de Verdiere's formulas  for Schrödinger operators with polynomial  magnetic and electric fields, Annales de l'Institut Fourier, vol. 56, no. 6 (2006), pp. 1827--1901. http://www.eco.utexas.edu/~leven/beyond6.pdf

S. Levendorskiĭ, Spectral properties of Schrödinger operators with irregular magnetic potentials, for a spin 1/2 particle. J. Math. Anal. Appl. 216 (1997), no. 1, 48--68.

2. Absolute continuity of spectrum of wide classes of operators with periodic coefficients

• P.Kuchment, S. Levendorskiĭ, On the structure of spectra of periodic elliptic operators, Trans. AMS, 354 (2002), 537-569   pdf file

3. Dicsrete spectrum in a gap of the absolute continuous spectrum

• S. Levendorskiĭ, Asymptotic formulae with remainder estimates for eigenvalue branches of the Schrödinger operator $H-\lambda W$ in a gap of $H$. Trans. Amer. Math. Soc. 351 (1999), no. 3, 857--899.
• S. Levendorskiĭ, Floquet's theory for the Schrödinger operator with perturbed periodic potential, and exact spectral asymptotics. (Russian) Dokl. Akad. Nauk 355 (1997), no. 1, 21--24.
• R. Hempel, S. Levendorskiĭ,On eigenvalues in gaps for perturbed magnetic Schrödinger operators.J. Math. Phys. 39 (1998), no. 1, 63--78.
• S. Levendorskiĭ,Acoustic waves in perturbed periodic layer: a limiting absorption principle. Asymptot. Anal. 16 (1998), no. 1, 15--24.

• S. Boyarchenko, S. Levendorskiĭ, Precise spectral asymptotics for perturbed magnetic Schrödinger operator. J. Math. Pures Appl. (9) 76 (1997), no. 3, 211--236.

• S.  Boyarchenko, S. Levendorskiĭ, An asymptotic formula for the number of eigenvalue branches of a divergence form operator $A+\lambda B$ in a spectral gap of $A$. Comm. Partial Differential Equations 22 (1997), no. 11-12, 1771--1786.
• S. Levendorskiĭ, Magnetic Floquet theory and spectral asymptotics for Schrödinger operators. (Russian) Funktsional. Anal. i Prilozhen. 30 (1996), no. 4,77--80 translation in Funct. Anal. Appl. 30 (1996), no. 4, 282--285 (1997)
• S. Levendorskiĭ, The asymptotics for the number of eigenvalue branches for the magnetic Schrödinger operator $H-\lambda W$ in a gap of $H$. Math. Z. 223 (1996), no. 4, 609--625.
• S. Levendorskiĭ,  Lower bounds for the number of eigenvalue branches for the Schrödinger operator $H-\lambda W$ in a gap of $H$: the case of indefinite $W$. Comm. Partial Differential Equations 20 (1995), no. 5-6, 827--854.

4. Spectral asymptotics of operators with discrete spectrum and generalizations of the classical Weyl formula: various classes of PDO, degenerate elliptic operators, hypoelliptic operators, boundary problems for these operators and several operators in mathematical physics and mechanics.

Main publications:

• S. Levendorskiĭ, Degenerate elliptic equations. Mathematics and its Applications, 258. Kluwer Academic Publishers Group, Dordrecht, 1993, xii+431 pp.
• S. Levendorskiĭ Asymptotic distribution of eigenvalues of differential operators. Translated from the Russian. Mathematics and its Applications (Soviet Series), 53. Kluwer Academic Publishers Group, Dordrecht, 1990, xviii+279 pp.
• S. Levendorskiĭ, The asymptotics of the spectrum of operators that are elliptic in the sense of Douglis-Nirenberg. (Russian) Trudy Moskov. Mat. Obshch. 52 (1989), 34-57, 247 translation in Trans. Moscow Math. Soc. 1990, , 33--57 (1991)
• S. Levendorskiĭ, Nonclassical spectral asymptotics. (Russian) Uspekhi Mat. Nauk 43 (1988), no. 1(259),123--157, 247 translation in Russian Math. Surveys 43 (1988), no. 1, 149--192
• S. Levendorskiĭ, Asymptotic behavior of the spectrum of problems with constraints. (Russian) Mat. Sb. (N.S.) 129(171) (1986), no. 1, 73--89, 159.
• S. Levendorskiĭ, Spectral asymptotics of weakly elliptic systems of pseudodifferential operators. (Russian) Sibirsk. Mat. Zh. 27 (1986), no. 1, 111--122, 199.
• S. Levendorskiĭ, The approximate spectral projector method. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), no. 6, 1177--1228, 1342.
• S. Levendorskiĭ, Spectral asymptotics of nonlinear pencils and operators with nonempty essential spectrum. (Russian) Dokl. Akad. Nauk SSSR 272 (1983), no. 6, 1314--1317.
• S. Levendorskiĭ, Asymptotic distribution of eigenvalues. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 4, 810--852.
• S. Levendorskiĭ, A general method for investigating the asymptotic behavior of a spectrum and its application to the theory of shells. (Russian) Dokl. Akad. Nauk SSSR 266 (1982), no. 2, 276--280.
• S. Levendorskiĭ, Quasiclassical asymptotic behavior of eigenvalues of systems of pseudodifferential operators. (Russian) Dokl. Akad. Nauk SSSR 256 (1981), no. 1, 32--36.

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University of Leicester
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