# 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)
Completeness of eigenfunctions of Sturm-Liouville problems with transmission conditions. (English) Zbl 1210.34123

This work considers the discontinuous boundary value problem with the eigenvalue parameter appearing linearly in the boundary conditions for a second order ordinary differential equation:

$-{\left(a\left(x\right){u}^{\text{'}}\right)}^{\text{'}}+q\left(x\right)u=\lambda u,\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}x\in \left[-1,0\right)\cup \left(0,1\right]$
${\alpha }_{1}u\left(-1\right)+{\alpha }_{2}{u}^{\text{'}}\left(-1\right)=0$
${\beta }_{1}u\left(1\right)-{\beta }_{2}{u}^{\text{'}}\left(1\right)=\lambda \left({\beta }_{2}^{\text{'}}{u}^{\text{'}}\left(1\right)-{\beta }_{1}^{\text{'}}u\left(1\right)\right)$
$u\left(0+\right)-{\alpha }_{3}u\left(0-\right)-{\beta }_{3}{u}^{\text{'}}\left(0-\right)=0,$
${u}^{\text{'}}\left(0+\right)-{\alpha }_{4}u\left(0-\right)-{\beta }_{4}{u}^{\text{'}}\left(0-\right)=0,$

where $\lambda$ is a spectral parameter, $a\left(x\right)={a}_{1}^{2}$ for $-1\le x\le 0$ and $a\left(x\right)={a}_{2}^{1}$ for $0\le x\le 1$, $\phantom{\rule{4pt}{0ex}}{a}_{i}\ne 0$ $\left(i=1,2\right)$ are real numbers, $q\left(x\right)\in {L}_{1}\left(-1,1\right),$ ${\alpha }_{i},$ ${\beta }_{i},$ ${\beta }_{j}^{\text{'}}\in ℝ$ $\left(i=1,2,3,4,$ $j=1,2\right)$ and ${\alpha }_{3}{\beta }_{4}-{\alpha }_{4}{\beta }_{3}>0,$ ${\beta }_{1}^{\text{'}}{\beta }_{2}-{\beta }_{1}{\beta }_{2}^{\text{'}}>0,$ $\left|{\alpha }_{1}\right|+\left|{\alpha }_{2}\right|\ne 0·$

Firstly, the given boundary value problem is reduced to an linear operator $A$ in a special Hilbert space $H$. Then, it is shown that the operator $A$ is selfadjoint in the space $H$. It is proved that the operator $A$ has only point spectrum and the corresponding eigenfunctions form an orthonormal basis in $H$.

There are many studies about boundary problems with discontinuous coefficients and transmission conditions, completeness, minimality and basis property of the eigenfunctions of this boundary value problem [for example, see A. Gomilko and V. Pivovarcik, Math. Nachr. 245, 72–93 (2002; Zbl 1023.34023); V. Pivovarcik, Asymptotic Analysis., 26 (2001)]; R. Kh. Amirov, A. Z. Ozkan and B. Keskin, Integral Transforms Spec. Funct. 20, No. 7–8, 607–618 (2009; Zbl 1181.34019) and others].

##### MSC:
 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE) 47E05 Ordinary differential operators 34B08 Parameter dependent boundary value problems for ODE 34B24 Sturm-Liouville theory