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)
Spectral properties of non-selfadjoint difference operators. (English) Zbl 0992.39018
The authors consider the operator $L$ generated in $\ell^2({\Bbb Z})$ by the difference expression $(\ell y)_n=a_{n-1}y_{n-1}+b_ny_n+a_ny_{n+1}$, $n\in{\Bbb Z}$, where $\{a_n\}_{n\in{\Bbb Z}}$ and $\{b_n\}_{n\in{\Bbb Z}}$ are complex sequences. The spectrum, the spectral singularities, and the properties of the principal vectors corresponding to the spectral singularities of $L$ are investigated. The authors also study similar problems for the discrete Dirac operator generated in $\ell({\Bbb Z,\Bbb C}^2)$ by the system of the difference expression $$ \pmatrix \Delta y_n^{(2)}+p_ny_n^{(1)}\cr -\Delta y_{n-1}^{(1)}+q_ny_n^{(2)} \endpmatrix, $$ $n\in{\Bbb Z}$, where $\{p_n\}_{n\in{\Bbb Z}}$ and $\{q_n\}_{n\in{\Bbb Z}}$ are complex sequences.

MSC:
39A70Difference operators
39A12Discrete version of topics in analysis
34L05General spectral theory for OD operators
WorldCat.org
Full Text: DOI
References:
[1] Agarwal, R. P.; Wong, P. J. Y.: Advanced topics in difference equations. (1997) · Zbl 0878.39001
[2] Bairamov, E.; Çakar, Ö.; Çelebi, A. O.: Quadratic pencil of Schrödinger operators with spectral singularities: discrete spectrum and principal functions. J. math. Anal. appl. 216, 303-320 (1997) · Zbl 0892.34077
[3] Bairamov, E.; Çakar, Ö.; Krall, A. M.: Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition. J. differential equations 151, 252-267 (1999) · Zbl 0927.34063
[4] Bairamov, E.; Çakar, Ö.; Krall, A. M.: An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities. J. differential equations 151, 268-289 (1999) · Zbl 0945.34067
[5] Bairamov, E.; Çelebi, A. O.: Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators. Quart. J. Math. Oxford ser. (2) 50, 371-384 (1999) · Zbl 0945.47026
[6] Bairamov, E.; Çelebi, A. O.: Spectral properties of the Klein--Gordon s-wave equation with complex potential. Indian J. Pure appl. Math. 28, 813-824 (1997) · Zbl 0880.34088
[7] Bairamov, E.; Tunca, G. B.: Discrete spectrum and principal functions of non-selfadjoint differential operators. Czechoslovak math. J. 49, 689-700 (1999) · Zbl 1015.34073
[8] Berezanski, Yu.M.: Expansion in eigenfunctions of self-adjoint operators. (1968)
[9] Dolzhenko, E. P.: Boundary value uniqueness theorems for analytic functions. Math. notes 26, 437-442 (1979) · Zbl 0441.30044
[10] I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Jerusalem, 1965. · Zbl 0143.36505
[11] Guseinov, G. S.: The inverse problem of scattering theory for a second order difference equation on the whole axis. Soviet math. Dokl. 17, 1684-1688 (1976) · Zbl 0398.39002
[12] Lyance, V. E.: A differential operator with spectral singularities, I, II. (1967)
[13] Naimark, M. A.: Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second order on a semi-axis. (1960)
[14] Naimark, M. A.: Linear differential operators, II. (1968) · Zbl 0227.34020