On two spectral problems with the same characteristic equation (Q745275)

The boundary value problem \[ xu''(x)+u'(x)+\lambda x u(x)=0, \quad 0<a<x<b, \tag{1} \] \[ u(a)=0, \quad u'(b)-\lambda d u(b)=0 \tag{2} \] is considered. It is shown that there is a relation between the solution of the boundary value problem (1), (2) and the solution of the boundary value problem (3), (4): \[ xv''(x)+v'(x)+\left(\lambda x - \frac{1}{x}\right)v(x)=0, \quad 0<a<x<b, \;\tag{3} \] \[ av'(a)+v(a)=0, \quad dbv'(b)+(d+b)v(b)=0, \tag{4} \] where \(d\) is a nonzero real coefficient. The following two theorems are proven: Theorem. The system \(\left\{u_{n}(x)\right\}\), \(n=1,2, \dots l-1, l+1, \dots\), of eigenfunctions of problem (1), (2) without an arbitrary eigenfunction \(u_{l}(x)\) is a Riesz basis in the weighted space \(L_{2}(a,b)\). The biorthogonal system \(\left\{\Psi_{n}(x)\right\}\) is given by \[ \Psi_{n}(x)=\left(\int^{b}_{a} xu_{n}^{2}(x)dx+dbu_{n}^{2}(b)\right)^{-1}\left[u_{n}(x)-\frac{u_{n}(b)}{u_{l}(b)}u_{l}(x)\right]. \] Theorem. Let the function \(f(x)\) be zero at the point \(x=a\) and be smooth enough on \([a,b]\) to ensure that its derivative provides the uniform convergence of the Fourier series on \([a,b]\) in the orthonormal basis of problem (3), (4). Then the function \(f(x)\) can be expanded in the uniformly convergent Fourier series \[ f(x)=\sum^{\infty}_{n=1} \left(\int^{b}_{a}tu_{n}^{2}(t)dt+dbu_{n}^{2}(b)\right)^{-1}\left[\int^{b}_{a}tu_{n}(t)f(t)dt+dbu_{n}(b)f(b)\right]u_{n}(x) \] on \([a,b]\) in the system \(u_{n}(x)\), \(n=1,2,3, \dots\).