A priori bounds and existence of non-real eigenvalues of indefinite Sturm–Liouville problems

Abstract

Abstract. The present paper gives a priori bounds on the possible non-real eigenvalues of regular indefinite Sturm-Liouville problems and obtains sufficient conditions for such problems to admit non-real eigenvalues.Mathematics Subject Classification(2010): 34B24, 34L15, 47B50

On a Non-Definite Sturm-Liouville Problem in the Two-Turning Point Case—Analysis and Numerical Results

“…It was shown in [18] and [8] The summary of the results are shown in Table 1 and Table 2. Table 1 brings out the difference between the number of zeros of real and imaginary parts of the non-real eigenfunctions corresponding to non-real eigenvalues of the problem (8)- (9). The results in this table are complemented by the results shown in Figure 1 which shows that the number of zeros of the real and imaginary parts of the non-real eigenfunctions are either equal or differ by two.…”

“…Here we give a summary on the non-definite case, detailed literature can be found in the papers [3]- [9], etc, and the references there in. In the non-definite case the spectrum is discrete, always consists of a doubly infinite sequence of real eigenvalues, and has at most a finite and even number of non-real eigenvalues (necessarily occurring in complex conjugate pairs).…”

Estimates on the non-real eigenvalues of regular indefinite Sturm–Liouville problems

“…. Then w ∈ L 1 (0, 1 π ), but w / ∈ AC[0, 1 π ] (and hence the results in [15] do not apply here). In order to estimate the non-real eigenvalues of the equation (1.1) with some q ∈ L 1 (0, 1 π ) and some selfadjoint boundary conditions, we put g(x) := x 4 sin(1/x).…”

Bounds on Non‐Real Eigenvalues of Indefinite Sturm‐Liouville Problems

“…The first results of the desired form were obtained by the first and third author jointly with Trunk in [2] for a singular problem. Independently, the second and fourth author found a priori bounds in [7] for the regular case. Both contributions [2,7] consider special cases such as, e.g., only one turning point of the weight w. The regular problem was then solved almost completely in the even more recent paper [1].…”

“…Independently, the second and fourth author found a priori bounds in [7] for the regular case. Both contributions [2,7] consider special cases such as, e.g., only one turning point of the weight w. The regular problem was then solved almost completely in the even more recent paper [1]. In fact, a priori bounds on the non-real eigenvalues were obtained for all selfadjoint boundary conditions, all potentials q, and all weight functions w for which there exists an absolutely continuous function g ∈ H 1 (a, b) such that sgn(g) = sgn(w) a.e.…”