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)
The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential. (English) Zbl 1003.34075

It is wellknown that the spectral theory of operators L=d 2 dx 2 +q(x) with a complex 1-periodic potential can be reduced to the study of operators L t =L|[0,1], which are defined by the t-periodic boundary conditions y(1)=e it y(0) and y ' (1)=e it y ' (0). For the spectral expansion, the authors determine first the asymptotics of the eigenvalues and eigenfunctions of the operators L t , t0, π. Their key result is the estimate

ψ n,t (x)=expi(2πn+t)x+O1 n

on the eigenfunctions. This is used to show that ψ=ψ n,t 0 and the associated functions ψ n,t k =(L t -λ n )ψ n,t k-1 form a Riesz basis of L 2 [0,1] by constructing a biorthonormal system with eigenfunctions of L t ¯ . Using this result, a spectral representation of functions with compact support can be derived.

MSC:
34L10Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE)
34L30Nonlinear ordinary differential operators
34B30Special ODE (Mathieu, Hill, Bessel, etc.)