# 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)
Inverse problems for differential operators with singular boundary conditions. (English) Zbl 1090.34007

The authors study inverse spectral problems for differential operators generated by differential equations of the form

$-{\left({p}_{2}\left(t\right){z}^{\text{'}}\left(t\right)\right)}^{\text{'}}+{p}_{1}\left(t\right)z\left(t\right)=\lambda {p}_{0}\left(t\right)z\left(t\right)$

considered on a finite interval $t\in \left({t}_{0},{t}_{1}\right)$ and with the boundary conditions at the end point of the interval. The specific feature of the study is that the coefficients ${p}_{k}\left(t\right)$ are assumed to have singularities at ${t}_{0}$ and ${t}_{1}$, which requires the boundary conditions to be properly understood. In the paper, the case of the so-called regular singularities is treated, which is characterized by the fact that the Liouville transformation applied to the differential equation leads to the Sturm-Liouville equation on a finite interval with a potential having quadratic singularities at the end points.

The authors study three inverse problems, where the operator is to be recovered by using either the Weyl function or the spectral data or two spectra. The corresponding uniqueness theorems are proven, and relations among the different spectral characteristics are established. A constructive procedure for recovering the operator from the spectral data is given.

##### MSC:
 34A55 Inverse problems of ODE 34B24 Sturm-Liouville theory 34L40 Particular ordinary differential operators 47E05 Ordinary differential operators