Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit

Schmidt, Karl

doi:10.1090/S0002-9939-02-06679-0

Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit
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Proc. Amer. Math. Soc. 131 (2003), 1205-1214 Request permission

Abstract:

A perturbation decaying to $0$ at $\infty$ and not too irregular at $0$ introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues in a compact subset of a gap in the essential spectrum is given by a quasi-semiclassical asymptotic formula in the slow-decay limit, which for power-decaying perturbations is equivalent to the large-coupling limit. This asymptotic behaviour elucidates the origin of the dense point spectrum observed in spherically symmetric, radially periodic three-dimensional Dirac operators.

References

Stanley Alama, Percy A. Deift, and Rainer Hempel, Eigenvalue branches of the Schrödinger operator $H-\lambda W$ in a gap of $\sigma (H)$, Comm. Math. Phys. 121 (1989), no. 2, 291–321. MR 985401
M. Sh. Birman, Discrete spectrum in the gaps of the continuous one in the large-coupling-constant limit, Order, disorder and chaos in quantum systems (Dubna, 1989) Oper. Theory Adv. Appl., vol. 46, Birkhäuser, Basel, 1990, pp. 17–25. MR 1124649
M. Sh. Birman, Discrete spectrum in the gaps of a continuous one for perturbations with large coupling constant, Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–90) Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 57–73. MR 1306508
M. Sh. Birman, On a discrete spectrum in gaps of a second-order perturbed periodic operator, Funktsional. Anal. i Prilozhen. 25 (1991), no. 2, 89–92 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 2, 158–161. MR 1142222, DOI 10.1007/BF01079605
M. Sh. Birman, The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. I. Regular perturbations, Boundary value problems, Schrödinger operators, deformation quantization, Math. Top., vol. 8, Akademie Verlag, Berlin, 1995, pp. 334–352. MR 1389015
M. Sh. Birman, The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential, Algebra i Analiz 8 (1996), no. 1, 3–20 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 1, 1–14. MR 1392011
M. Sh. Birman, The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. II. Nonregular perturbations, Algebra i Analiz 9 (1997), no. 6, 62–89 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 6, 1073–1095. MR 1610239
Mikhail Sh. Birman and Ari Laptev, Discrete spectrum of the perturbed Dirac operator, Ark. Mat. 32 (1994), no. 1, 13–32. MR 1277918, DOI 10.1007/BF02559521
M. Sh. Birman and A. Laptev, The negative discrete spectrum of a two-dimensional Schrödinger operator, Comm. Pure Appl. Math. 49 (1996), no. 9, 967–997. MR 1399202, DOI 10.1002/(SICI)1097-0312(199609)49:9<967::AID-CPA3>3.3.CO;2-O
Mikhail Sh. Birman, Ari Laptev, and Michael Solomyak, The negative discrete spectrum of the operator $(-\Delta )^l-\alpha V$ in $L_2(\textbf {R}^d)$ for $d$ even and $2l\geq d$, Ark. Mat. 35 (1997), no. 1, 87–126. MR 1443037, DOI 10.1007/BF02559594
Brown B.M., Eastham M.S.P., Hinz A.M., Schmidt K.M., Distribution of eigenvalues in gaps of the essential spectrum of Sturm-Liouville operators — a numerical approach, J. Comp. Anal. Appl. (to appear).
C. Cancelier, P. Lévy-Bruhl, and J. Nourrigat, Remarks on the spectrum of Dirac operators, Acta Appl. Math. 45 (1996), no. 3, 349–364. MR 1420481, DOI 10.1007/BF00047028
Eastham M.S.P., The spectral theory of periodic differential equations, Scottish Academic Press, Edinburgh 1973.
Rainer Hempel, Ira Herbst, Andreas M. Hinz, and Hubert Kalf, Intervals of dense point spectrum for spherically symmetric Schrödinger operators of the type $-\Delta +\cos |x|$, J. London Math. Soc. (2) 43 (1991), no. 2, 295–304. MR 1111587, DOI 10.1112/jlms/s2-43.2.295
M. Klaus, On the point spectrum of Dirac operators, Helv. Phys. Acta 53 (1980), no. 3, 453–462 (1981). MR 611769
Ari Laptev, Asymptotics of the negative discrete spectrum of a class of Schrödinger operators with large coupling constant, Proc. Amer. Math. Soc. 119 (1993), no. 2, 481–488. MR 1149974, DOI 10.1090/S0002-9939-1993-1149974-0
Jürgen Moser, An example of a Schroedinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helv. 56 (1981), no. 2, 198–224. MR 630951, DOI 10.1007/BF02566210
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
F. S. Rofe-Beketov, Spectral analysis of the Hill operator and of its perturbations, Functional analysis, No. 9: Harmonic analysis on groups (Russian), Ul′janovsk. Gos. Ped. Inst., Ul′yanovsk, 1977, pp. 144–155 (Russian). MR 0493524
F. S. Rofe-Beketov, Generalization of the Prüfer transformation and the discrete spectrum in gaps of the continuous spectrum, Spectral theory of operators (Proc. Second All-Union Summer Math. School, Baku, 1975) “Èlm”, Baku, 1979, pp. 146–153 (Russian). MR 558545
Rofe-Beketov F.S., Spectrum perturbations, the Kneser-type constants and the effective masses of zones-type potentials, Sofia 1984, pp. 757–766.
F. S. Rofe-Beketov, Constants of Kneser type and effective masses for zone potentials, Dokl. Akad. Nauk SSSR 276 (1984), no. 2, 356–359 (Russian). MR 745043
Karl Michael Schmidt, On the essential spectrum of Dirac operators with spherically symmetric potentials, Math. Ann. 297 (1993), no. 1, 117–131. MR 1238410, DOI 10.1007/BF01459491
Karl Michael Schmidt, Dense point spectrum and absolutely continuous spectrum in spherically symmetric Dirac operators, Forum Math. 7 (1995), no. 4, 459–475. MR 1337149, DOI 10.1515/form.1995.7.459
Karl Michael Schmidt, Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm-Liouville operators, Comm. Math. Phys. 211 (2000), no. 2, 465–485. MR 1754525, DOI 10.1007/s002200050822
A. V. Sobolev, Weyl asymptotics for the discrete spectrum of the perturbed Hill operator, Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–90) Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 159–178. MR 1306512
Joachim Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, Berlin, 1987. MR 923320, DOI 10.1007/BFb0077960