On the completeness of the system of Weber functions. (English) Zbl 1513.34355
Summary: Weber functions \(D_{\frac{\lambda_n-1}{2}} \left(\sqrt{2}x\right)\), \(n= 0,1,2, \dots\), are considered, where \(\lambda_n\) is the eigenvalue of the perturbed harmonic oscillator on the semi-axis with finite potential and with the Dirichlet boundary condition at zero. The completeness in the space \(L_2 (0,\infty)\) of a system of functions \(\left\{D_{\frac{\lambda_n-1}{2}} \left(\sqrt{2}x \right)\right\}_{n=0}^\infty\) is proved.
MSC:
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
Keywords:
Weber function; perturbed harmonic oscillator; eigenvalues; completeness of a system of functionsReferences:
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