Abstract
We consider a spectral problem for a class of singular Sturm-Liouville operators on the unit interval with explicit singularity $2/x$ - $2/x^{2}$ , related to the Schrodinger operator with radially symmetric potential. In particular, we give the asymptotic behavior of the eigenvalues of the hydrogen atom equation.
Highlights
1 Introduction The distribution of eigenvalues in differential operator’s spectral theory has an important place. This classic issue was first examined in a finite interval for second order operators in the th century by Sturm and Liouville
The distribution of eigenvalues of the operators with a discrete spectrum defined in the whole of space for quantum mechanics has great importance
The formula for the distribution of the eigenvalues of the single-dimensional Sturm operator defined in the whole of the straight-line axis with increasing potential at infinity was given by Titchmarsh in [ ]
Summary
The distribution of eigenvalues in differential operator’s spectral theory has an important place This classic issue was first examined in a finite interval for second order operators in the th century by Sturm and Liouville. Levitan and Gasymov improved the Titchmarsh method and found important asymptotic formulas for the eigenvalues of different differential operators [ , ]. The second method that is related with the resolvent of the operator in question was suggested by Carleman [ ] Another important method for examining the asymptotic of the eigenvalues in singular condition was suggested by Fedoryuk [ ]. This method is very useful in that it ensures that the distribution of the eigenvalues of the operators with partial derivation are such that the coefficients are analytic functions. The inverse problem was examined in Panakhov and Yilmazer’s papers [ , ]
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