On the equiconvergence rate with the Fourier integral of the spectral expansion associated with the self-adjoint extension of the Sturm-Liouville operator with uniformly locally integrable potential
In this paper, we consider properties of the spectrum of a Sturm-Liouville<br />operator on time scales. We will prove that the regular symmetric<br />Sturm-Liouville operator is semi-bounded from below. We will also give some<br />conditions for the self-adjoint operator associated with the singular<br />Sturm-Liouville expression to have a discrete spectrum. Finally, we will<br />investigate the continuous spectrum of this operator.
Let H0 be the mth power (m a positive integer) of the self-adjoint operator defined in the Hilbert space L2(0, π) by the differential operator — (d2/dx2) and the boundary conditions u(0) = u(π) = 0. The eigenvalues of H0 are μn = n2m and the corresponding eigenfunctions are ϕn = (2/π)1/2 sin nx, n = 1 , 2 , . . ..Let p be a (2m — 2)-times continuously differentiate real valued function defined over the interval [0, π] satisfying the conditions p(j)(0) = p(j)(π) = 0 for j odd and less than 2m — 4.
An inverse spectral problem for the Sturm-Liouville operator with a singular potential from the class $W_2^{-1}$ is solved by the method of spectral mappings. We prove the uniqueness theorem, develop a constructive algorithm for solution and obtain necessary and sufficient conditions of solvability for the inverse problem in the self-adjoint and the non-self-adjoint cases.
Spectral analysis of singular Sturm-Liouville operators on time scales