Left-definite Problems Research Articles

Eigenvalue estimates for singular left-definite Sturm–Liouville operators

The spectral properties of a singular left-definite Sturm–Liouville operator JA are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart A which is obtained by substituting the indefinite weight function by its absolute value. The spectrum of the J -selfadjoint operator JA is real and it follows that an interval (a,b)\subset\mathbb R^+ is a gap in the essential spectrum of A if and only if both intervals (-b,-a) and (a,b) are gaps in the essential spectrum of the J -selfadjoint operator JA . As one of the main results it is shown that the number of eigenvalues of JA in (-b,-a) \cup (a,b) differs at most by three of the number of eigenvalues of A in the gap (a,b) ; as a byproduct results on the accumulation of eigenvalues of singular left-definite Sturm-Liouville operators are obtained. Furthermore, left-definite problems with symmetric and periodic coefficients are treated, and several examples are included to illustrate the general results.

Singular left-definite Sturm–Liouville problems

We study singular left-definite Sturm–Liouville problems with an indefinite weight function. The existence of eigenvalues is established based on the existence of eigenvalues of corresponding right-definite problems. Furthermore, for each singular left-definite problem with limit-circle non-oscillatory endpoints we construct a regular left-definite problem with the same eigenvalues and use it to obtain properties of eigenvalues and eigenfunctions. Inequalities among eigenvalues recently established for regular left-definite problems are extended to the singular case.

Left-Definite Sturm–Liouville Problems

Left-definite regular self-adjoint Sturm–Liouville problems, with either separated or coupled boundary conditions, are studied. We give an elementary proof of the existence of eigenvalues for these problems. For any fixed equation, we establish a sequence of inequalities among the eigenvalues for different boundary conditions and estimate the range of each eigenvalue as a function on the space of boundary conditions. Some of our results here yield an algorithm for numerically computing the eigenvalues of a left-definite problem with an arbitrary coupled boundary condition. Our inequalities imply that the well-known asymptotic formula for the eigenvalues in the separated case also holds in the coupled case. Moreover, we study the continuous and differentiable dependence of the eigenvalues of the general left-definite problem on all the parameters in its differential equation and boundary condition.

Differential Operators and the Laguerre Type Polynomials

In 1940, all fourth-order differential equations which have a sequence of orthogonal polynomial eigenfunctions were classified by H. L. Krall, up to a linear change of variable. One of these equations was subsequently named the Laguerre type equation and various properties of the orthogonal polynomial solutions and the right-definite boundary value problem were studied by A. M. Krall in 1981. In this paper, the Laguerre type expression is further studied in the right-definite setting and the appropriate left-definite problem associated with the fourth-order Laguerre type differential expression is discussed in detail.