zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Singular Dirac systems and Sturm-Liouville problems nonlinear in the spectral parameter. (English) Zbl 1012.34082
Here, the Dirac operator $$ H=-i\alpha \nabla +\beta+V$$ acting in $L^2(\bbfR^3)^4$ with a spherically symmetric potential $V$ is studied. Under certain additional assumptions on $V$, the essential spectrum satisfies $\sigma_{ess}(H)= \bbfR\backslash (-1,1)$. The authors investigate whether the endpoints $-1$ and $1$ of the gap $(-1,1)$ are accumulation points of discrete eigenvalues.

34L40Particular ordinary differential operators
34L05General spectral theory for OD operators
34B07Linear boundary value problems with nonlinear dependence
34B16Singular nonlinear boundary value problems for ODE
34B24Sturm-Liouville theory
34B40Boundary value problems for ODE on infinite intervals
Full Text: DOI
[1] Ahlbrandt, C. D.: Principal and antiprincipal solutions of selfadjoint differential systems and their reciprocals. Rocky mountain J. Math. 2, 169-182 (1972) · Zbl 0236.34031
[2] Eastham, M. S. P.: The asymptotic solution of linear differential systems: applications of the Levinson theorem. (1989) · Zbl 0674.34045
[3] Griesemer, M.; Lutgen, J. P.: Accumulation of discrete eigenvalues of the radial Dirac operator. J. funct. Anal. 162, 120-134 (1999) · Zbl 0926.34074
[4] Hartman, P.: Ordinary differential equations. (1964) · Zbl 0125.32102
[5] Lutgen, J. P.: Eigenvalue accumulation for singular Sturm--Liouville problems nonlinear in the spectral parameter. J. differential equations 159, 515-542 (1999) · Zbl 0951.34058
[6] Mennicken, R.; Schmid, H.; Shkalikov, A. A.: On the eigenvalue accumulation of Sturm--Liouville problems depending nonlinearly on the spectral parameter. Math. nachr. 189, 157-170 (1998) · Zbl 0892.34019
[7] Prevatt, T. W.: Singular boundary value problems for linear Hamiltonian systems depending on a parameter. J. differential equations 20, 1-17 (1976) · Zbl 0347.34019
[8] Reid, W. T.: A continuity property of principal solutions of linear Hamiltonian differential systems. Scripta math. 29, 337-350 (1973) · Zbl 0267.34030
[9] Reid, W. T.: Sturmian theory for ordinary differential equations. (1980) · Zbl 0459.34001
[10] Schmid, H.: Das häufungsverhalten der eigenwerte bei randeigenwertproblemen zu störungen regulär-singulärer differentialgleichungen. (1998)
[11] Thaller, B.: The Dirac equation. (1992) · Zbl 0765.47023
[12] Vogelsang, V.: Remark on essential selfadjointness of Dirac operators with Coulomb potentials. Math. Z. 196, 517-521 (1987) · Zbl 0617.35127
[13] Weidmann, J.: Linear operators in Hilbert spaces. (1980) · Zbl 0434.47001
[14] Weidmann, J.: Spectral theory of ordinary differential operators. (1987) · Zbl 0647.47052