Structure of the spectrum of the Schrodinger operator with magnetic field in a strip and infinite-gap potentials

V. A. Geiler   ; M. M. Senatorov

Sbornik. Mathematics, Tome 188 (1997) no. 5, pp. 657-669

Résumé

The Sturm–Liouville operator $H=-d^2/dx^2+V(x+p)$ on an interval $[a,b]$ with zero boundary conditions is considered; here $V$ is a strictly convex function of class $C^2$ on the real line $\mathbb R$ and $p$ is a numerical parameter. The dependence of the eigenvalues of $H$ on $p$ is studied. The spectral analysis of the Schrödinger operator with magnetic field in a strip with Dirichlet boundary conditions on the boundary of the strip reduces to this problem. As a consequence of the main result the following theorem is obtained. Let $V_1$ be the restriction of $V$ to the interval $[a,b)$ and let $u$ be the periodic extension of $V_1$ on the entire axis (with period $b-a$). Then all the gaps in the spectrum of the Schrödinger operator $-d^2/dx^2+u(x)$ are non-trivial.

Bibliographie

[1] Tsikon Kh., Freze R., Kirsh V., Saimon B., Operatory Shredingera s prilozheniyami k kvantovoi mekhanike i globalnoi geometrii, Mir, M., 1990 | MR

[2] von Klitzing K., Dorda G., Pepper M., “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance”, Phys. Rev. Lett., 45:6 (1980), 494–497 | DOI

[3] Kvantovyi effekt Kholla, Sb. statei, eds. A. Ya. Shik, Yu. V. Shmartsev, Mir, M., 1986

[4] Kvantovyi effekt Kholla, eds. R. Prendzh, S. Girvin, Mir, M., 1989

[5] Novikov S. P., “Magnitno-blokhovskie funktsii i vektornye rassloeniya. Tipichnye zakony dispersii i ikh kvantovye chisla”, Dokl. AN SSSR, 257:3 (1981), 538–543 | MR | Zbl

[6] Bellissard J., “$\mathrm C^*$-algebras in solid state physics. 2D-electrons in a uniform magnetic field”, London Math. Soc. Lecture Note Ser., no. 136, 1988, 49–76 | MR | Zbl

[7] Bellissard J., van Elst A., Schulz-Baldes H., “The noncommutative geometry of the quantum Hall effect”, J. Math. Phys., 35:10 (1994), 5373–5451 | DOI | MR | Zbl

[8] Halperin B., “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential”, Phys. Rev. B, 25:4 (1982), 2185–2190 | DOI

[9] Ando T., “Edge states in quantum wires in high magnetic fields”, Phys. Rev. B, 42:9 (1990), 5626–5634 | DOI | MR

[10] Büttiker M., “Transmission probabilities and the quantum Hall effect”, Surf. Sci., 229:1 (1990), 201–208 | DOI

[11] Isihara A., Ebina K., “The density of states of narrow two-dimensional electron system in a magnetic field”, J. Phys. C, 33:30 (1988), L1079–L1083 | DOI

[12] Středa P., Kučera J., MacDonald A. M., “Edge states, transmission matrices, and the quantum Hall resistance”, Phys. Rev. Lett., 59:17 (1987), 1973–1975 | DOI

[13] Kinaret J. M., Lee P. A., “Conductance of a disorder narrow wire in a strong magnetic field”, Phys. Rev. B, 43:5 (1991), 3847–3855 | DOI

[14] Viehweger O., Pook W., Janßen M., Hajdu J., “Note on the quantum Hall effect in cylinder geometry”, Z. Phys. B, 78:1 (1990), 11–16 | DOI

[15] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, T. 4, Mir, M., 1982

[16] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR

[17] Levitan B. M., Sargsyan I. S., Operatory Shturma–Liuvillya i Diraka, Nauka, M., 1988 | MR | Zbl

[18] Titchmarsh E. Ch., Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, T. I, IL, M., 1961

[19] Kato K., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl