zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On perturbations of quasiperiodic Schrödinger operators. (English) Zbl 1214.34077

Using relative oscillation theory and the reducible result of Eliasson studied perturbations of quasiperiodic Schrödinger operators.

Let 𝐓 d be a d-dimensional torus, where 𝐓= /(2π). Let Q:𝐓 d be a real analytic function. The Schrödinger operator on L(1,) is given by

H 0 =-d 2 dx 2 +q 0 (x),q 0 (x)=Q(ωx),

where ω𝐓 d is fixed. The author investigates decaying perturbations of the quasiperiodic operator H 0 . That is, he considers the operator

H 1 =-d 2 dx 2 +q 1 (x),q 1 (x)=q 0 (x)+Δq(x)

for some function Δq. Assume that Δq(x)0 as x, and let E, EE 0 , be a boundary point of the essential spectrum of H 0 (E 0 is a certain constant). Then

σ ess (H 1 )=σ ess (H 0 )= n G n

for open sets G n and there exists a constant K=K(E) such that E is an accumulation point of eigenvalues of H 1 if

lim sup x KΔq(x)x 2 <-1 4

and E is not an accumulation point of eigenvalues of H 1

lim inf x KΔq(x)x 2 >-1 4·

Also, similar results are given for the operator H μ γ H 0 +μ x γ , where 0<γ2.

MSC:
34L05General spectral theory for OD operators
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34B24Sturm-Liouville theory
34L20Asymptotic distribution of eigenvalues for OD operators
34L40Particular ordinary differential operators
47E05Ordinary differential operators
References:
[1]Deift, P.; Killip, R.: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Comm. math. Phys. 203, 341-347 (1999) · Zbl 0934.34075 · doi:10.1007/s002200050615
[2]Denisov, S.; Kiselev, A.: Spectral properties of Schrödinger operators with decaying potentials, Proc. sympos. Pure math. 76, 565-589 (2007) · Zbl 1132.35352
[3]Eliasson, L. H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. math. Phys. 146, 447-482 (1992) · Zbl 0753.34055 · doi:10.1007/BF02097013
[4]Johnson, R.; Moser, J.: The rotation number for almost periodic potentials, Comm. math. Phys. 84, 403-438 (1982) · Zbl 0497.35026 · doi:10.1007/BF01208484
[5]Killip, R.: Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum, Int. math. Res. not. IMRN 38, 2029-2061 (2002) · Zbl 1021.34071 · doi:10.1155/S1073792802204250
[6]Kiselev, A.; Last, Y.; Simon, B.: Stability of singular spectral types under decaying perturbations, J. funct. Anal. 198, 1-27 (2003) · Zbl 1029.34073 · doi:10.1016/S0022-1236(02)00053-8
[7]Kirsch, W.; Simon, B.: Corrections to the classical behavior of the number of bound states of Schrödinger operators, Ann. physics 183, 122-130 (1988) · Zbl 0646.35019 · doi:10.1016/0003-4916(88)90248-5
[8]Kholkin, A. M.; Rofe-Beketov, F. S.: Spectral analysis of differential operators, World sci. Monogr. ser. Math. 7 (2005)
[9]Kneser, A.: Untersuchungen über die reellen nullstellen der integrale linearer differentialgleichungen, Math. ann. 42, 409-435 (1893) · Zbl 25.0522.01 · doi:http://jfm.sub.uni-goettingen.de/JFM/digit.php?an=JFM+25.0522.01
[10]Krüger, H.; Teschl, G.: Relative oscillation theory, zeros of the wronskians, and the spectral shift function, Comm. math. Phys. 287, No. 2, 613-640 (2009) · Zbl 1186.47009 · doi:10.1007/s00220-008-0600-8
[11]Krüger, H.; Teschl, G.: Relative oscillation theory for Sturm-Liouville operators extended, J. funct. Anal. 254, No. 6, 1702-1720 (2008) · Zbl 1144.34014 · doi:10.1016/j.jfa.2007.10.007
[12]Krüger, H.; Teschl, G.: Effective Prüfer angles and relative oscillation criteria, J. differential equations 245, 3823-3848 (2008) · Zbl 1167.34009 · doi:10.1016/j.jde.2008.06.004
[13]Moser, J.; Pöschel, J.: An extension of a result by dinaburg and Sinai on quasi-periodic potentials, Comment. math. Helv. 59, 39-85 (1984) · Zbl 0533.34023 · doi:10.1007/BF02566337
[14]Reed, M.; Simon, B.: Methods of modern mathematical physics, analysis of operators, (1978)
[15]Schmidt, K. M.: Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm-Liouville operators, Comm. math. Phys. 211, 465-485 (2000) · Zbl 0953.34069 · doi:10.1007/s002200050822