Abstract
We study the asymptotic behaviour ofN(α)—the number of negative eigenvalues of the operator (-τ)l -αV inL 2(R d) for an evend and2l≥d. This is the only case where the previously known results were far from being complete. In order to describe our results we introduce an auxiliary ordinary differential operator (system) on the semiaxis. Depending on the spectral properties of this operator we can distinguish between three cases whereN(α) is of the Weyl-type,N(α) is of the Weyl-order but not the Weyl-type coefficient and finally whereN(α)=O(αq) withq>d/2l.
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[B]Birman, M. Sh., On the spectrum of singular boundary-value problems,Mat. Sb. 55 (1961), 125–174 (Russian). English transl.: inEleven Papers on Analysis, Amer. Math. Soc. Transl.53, pp. 23–80, Amer. Math. Soc., Providence, R. I., 1966.
[BL]Birman, M. Sh. andLaptev, A., The negative discrete spectrum of a two-dimensional Schrödinger operator,Comm. Pure Appl. Math. 49 (1996), 967–997.
[BLS]Birman, M. Sh., Laptev, A. andSolomyak, M., On the eigenvalue behaviour for a class of differential operators on the semiaxis, to appear inMath. Nachr.
[BS1]Birman, M. Sh. andSolomyak, M. Z., The leading term in the asymptotic spectral formula for “non-smooth” elliptic problems,Funktsional Anal. i Prilozhen. 4 (1970), 1–13 (Russian). English transl.:Functional Anal. Appl. 4 (1970), 265–275.
[BS2]Birman, M. Sh. andSolomyak, M. Z., Spectral asymptotics of nonsmooth elliptic operators. I,Trudy Moskov. Mat. Obshch. 27 (1972), 3–52 (Russian). English transl.:Trans. Moscow Math. Soc. 27 (1972), 1–52.
[BS3]Birman, M. Sh. andSolomyak, M. Z., Quantitive analysis in Sobolev imbedding theorems and applications to spectral theory, inTenth Mathematical School (Mitropol'skiî, Yu. O. and Shestopal, A. F., eds.), pp. 5–189, Izdanie Inst. Mat. Akad. Nauk Ukrain. SSSR, Kiev, 1974 (Russian). English transl.: inQualitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory, Amer. Math. Soc. Transl.114, pp. 1–132, Amer. Math. Soc., Providence, R. I., 1980.
[BS4]Birman, M. Sh. andSolomyak, M. Z.,Spectral Theory of Selfadjoint Operators in Hilbert Space, Leningrad Univ., Leningrad, 1980 (Russian). English transl.: D. Reidel Publ. Co., Dordrecht, 1987.
[BS5]Birman, M. Sh. andSolomyak, M. Z., Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, inEstimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Birman, M. Sh., ed.), Adv. Soviet Math.7, pp. 1–55, Amer. Math. Soc., Providence, R. I., 1991.
[BS6]Birman, M. Sh. andSolomyak, M. Z., Schrödinger operator. Estimates of the number of the bound states as a function-theoretic problem, Lectures of the XIV School on Operator Theory on Functional Spaces I, Novgorod State Pedagogical Institute, Novgorod, 1990 (Russian). English transl.: inSpectral Theory of Operators (Gindikin, S. G., ed.), Amer. Math. Soc. Transl.150, pp. 1–54, Amer. Math. Soc., Providence, R. I., 1992.
[C]Cwikel, M., Weak type estimates for singular values and the number of bound states of Schrödinger operators,Ann. of Math. 106 (1977), 93–100.
[EK]Egorov, Yu. V. andKondrat'ev, V. A., On the negative spectrum of an elliptic operator,Mat. Sb. 181:2 (1990), 147–166 (Russian). English transl.:Math. USSR-Sb. 69 (1991), 155–177.
[JMS]Jaksić, V., Molcanov, S. andSimon, B., Eigenvalue asymptotics of the Neumann laplacian of regions and manifolds with cusps,J. Funct. Anal. 106 (1992), 59–79.
[KR]Krasnosel'skii, M. A. andRutickii, Ya. B.,Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow, 1958 (Russian). English transl.: P. Nordhoff, Groningen, 1961.
[L]Lieb, E. H., Bounds on the eigenvalues of the Laplace and Schrödinger operators,Bull. Amer. Math. Soc. 82 (1976), 751–753.
[NS]Naimark, K. andSolomyak, M., Regular and patological eigenvalue behaviour for the equation−lu=Vu on the semiaxis, Preprint.
[RS]Reed, M. andSimon, B.,Methods of Modern Mathematical Physics, 4, Academic Press, New York-San Francisco-London, 1978.
[R1]Rosenblum, G. V., The distribution of the discrete spectrum for singular differential operators,Dokl. Akad. Nauk SSSR 202 (1972), 1012–1015 (Russian). English transl.:Soviet Math. Dokl. 13 (1972).
[R2]Rosenblum, G. V., Distribution of the discrete spectrum of singular differential operators,Izv. Vyssh. Uchebn. Zaved. Mat. 1 (1976), 75–86 (Russian). English transl.:Soviet Math. (Iz. VUZ) 20 (1976), 63–71.
[S1]Solomyak, M., A remark on the Hardy inequalities,Integral Equations Operator Theory 19 (1994), 120–124.
[S2]Solomyak, M., Piecewise-polynomial approximation of functions fromH l((0, 1)d),2l=d, and applications to the spectral theory of Schrödinger operator,Israel J. Math. 86 (1994), 253–276.
[S3]Solomyak, M., Spectral problems related to the critical exponent in the Sobolev embedding theorem,Proc. London Math. Soc. 71 (1995), 53–75.
[SW]Stein, E. M. andWeiss, G.,Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N. J., 1971.
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Birman, M.S., Laptev, A. & Solomyak, M. The negative discrete spectrum of the operator (-τ)l -αV inL 2(R d) ford even and 2l≥dinL 2(R d) ford even and 2l≥d . Ark. Mat. 35, 87–126 (1997). https://doi.org/10.1007/BF02559594
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DOI: https://doi.org/10.1007/BF02559594