Geodesic
Asymptotics of the eigenvalues of the Schrödinger operator
Sbornik. Mathematics, Tome 22 (1974) no. 3, pp. 349-371 
G. V. Rozenblum. Asymptotics of the eigenvalues of the Schrödinger operator. Sbornik. Mathematics, Tome 22 (1974) no. 3, pp. 349-371. http://geodesic.mathdoc.fr/item/SM_1974_22_3_a1/
@article{SM_1974_22_3_a1,
     author = {G. V. Rozenblum},
     title = {Asymptotics of the eigenvalues of the {Schr\"odinger} operator},
     journal = {Sbornik. Mathematics},
     pages = {349--371},
     year = {1974},
     volume = {22},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1974_22_3_a1/}
}
TY  - JOUR
AU  - G. V. Rozenblum
TI  - Asymptotics of the eigenvalues of the Schrödinger operator
JO  - Sbornik. Mathematics
PY  - 1974
SP  - 349
EP  - 371
VL  - 22
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1974_22_3_a1/
LA  - en
ID  - SM_1974_22_3_a1
ER  - 
%0 Journal Article
%A G. V. Rozenblum
%T Asymptotics of the eigenvalues of the Schrödinger operator
%J Sbornik. Mathematics
%D 1974
%P 349-371
%V 22
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1974_22_3_a1/
%G en
%F SM_1974_22_3_a1

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

We examine the selfadjoint operator $H=-\Delta+V$ in $L_2(\mathbf R^m)$. We assume that the potential $V(x)\geqslant1$ tends to $+\infty$ as $|x|\to\infty$. Under these conditions the spectrum of $H$ is discrete. In the paper the well-known asymptotic formula \begin{equation} N(\lambda,H)\sim\gamma_m\int(\lambda-V(x))_+^{m/2}\,dx,\qquad\lambda\to\infty, \tag{\ast} \end{equation} for the distribution function of the eigenvalues is justified under very weak assumptions on $V$, namely the following conditions: 1) $\sigma(2\lambda)\leqslant c\sigma(\lambda)$, where $\sigma(\lambda)=\operatorname{mes}\{x:V(x)<\lambda\}$; 2) $V(x)\leqslant cV(y)$ almost everywhere when $|x-y|<1$; 3) there exist a continuous function $\eta(t)\geqslant0$, $0\leqslant t<1$, $\eta(0)=0$, and an index $\beta\in[0,1/2)$ such that $$ \int_{|x-y|\leqslant1,\,|x+z-y|\leqslant1}|V(x+z)-V(x)|\,dx<\eta(|z|)|z|^{2\beta}V(y)^{1+\beta} $$ for any $y\in\mathbf R^m$, $z\in\mathbf R^m$, $|z|<1$. Bibliography: 12 titles.

[1] J. de Wet, F. Mandle, “On the asymptotic distribution of eigenvalues”, Proc. Roy. Soc., A 200 (1950), 572–580 | MR | Zbl

[2] E. Tichmarsh, “On the asymptotic distribution of eigenvalues”, Quart. J. Math., 5 (1954), 228–240 | DOI | MR

[3] B. M. Levitan, “Ob asimptoticheskom povedenii funktsii Grina i razlozhenii po sobstvennym funktsiyam uravneniya Shredingera”, Matem. sb., 41 (83) (1957), 439–458 | MR | Zbl

[4] A. G. Kostyuchenko, “Asimptoticheskoe raspredelenie sobstvennykh znachenii ellipticheskikh operatorov”, DAN SSSR, 158:1 (1964), 41–44 | Zbl

[5] M. Sh. Birman, M. Z. Solomyak, “O glavnom chlene spektralnoi asimptotiki dlya negladkikh ellipticheskikh zadach”, Funkts. analiz, 4:4 (1970), 1–13 | MR | Zbl

[6] M. Sh. Birman, M. Z. Solomyak, “Spektralnaya asimptotika dlya negladkikh ellipticheskikh uravnenii. I; II”, Trudy Mosk. matem. ob-va, 27 (1972), 3–52 ; 28 (1973), 3–34 | MR | Zbl | MR | Zbl

[7] M. Sh. Birman, V. V. Borzov, “Ob asimptotike diskretnogo spektra nekotorykh singulyarnykh differentsialnykh operatorov”, Problemy matem. fiziki, no. 5, izd-vo LGU, Leningrad, 1971, 24–38 | MR

[8] G. V. Rozenblyum, “Raspredelenie diskretnogo spektra singulyarnykh differentsialnykh operatorov”, DAN SSSR, 202:5 (1972), 1012–1015

[9] W. Everitt, G. Giertz, “Some properties of domains of certain differential operators”, Proc. London. Math. Soc., 23:2 (1971), 301–324 | DOI | MR | Zbl

[10] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, izd-vo «Nauka», Moskva, 1965 | MR

[11] T. Kato, “Schrödinger operators with singular potentials”, Israel J. Math., 13:1–2 (1972), 135–148 | DOI | MR

[12] M. Sh. Birman, “O spektre singulyarnykh granichnykh zadach”, Matem. sb., 55 (97) (1961), 125–174 | MR | Zbl