We present the extension of the successful Constant Perturbation Method (CPM) for Schrödinger problems to the more general class of Sturm-Liouville eigenvalue problems. Whereas the orginal CPM can only be applied to Sturm-Liouville problems after a Liouville transformation, the more general CPM presented here solves the Sturm-Liouville problem directly. This enlarges the range of applicability ...

We present a new necessary condition for similarity of indefinite Sturm-Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl-Titchmarsh m-functions. Also we obtain necessary conditions for regularity of the critical points 0 and∞ of J-nonnegative Sturm-Liouville operators. Using this result, we construct several examples of operators with the singular cri...

In this paper, we investigate infinite product representation of the solution of a Sturm- Liouville equation with an indefinite weight function which has two zeros and/or singularities in a finite interval. First, by using of the asymptotic estimates provided in [W. Eberhard, G. Freiling, K. Wilcken-Stoeber, Indefinite eigenvalue problems with several singular points and turning points, Math. N...

In this work, we study the following conformable fractional Sturm--Liouville
 problem
 \[
 l[y]=-T_{\alpha }(p(t)T_{\alpha }y(t))+q(t)y(t),
 \]
 where $t\in \lbrack 0,\infty ),$ real-valued functions $p$ and $q$
 satisfy conditions:
 \begin{array}{cc}
 (i) & q\in L_{\alpha }^{2}[0,\infty ), \\ 
 (ii) p\ \text{is\ absolutely\ continuous\ on}\ [0,\...

The inverse Sturm-Liouville problem finds its applications in the identification of mechanical properties and/or geometrical configurations of a vibrating continuous medium; however, this problem is hard to solve, either theoretically or numerically. Previously, Liu (2008a) has constructed a Lie-group shooting method to determine the eigenvalues, and the corresponding eigenfunctions, for the di...

We consider a semilinear parabolic equation of the form ut = uxx + f(u, ux) defined on the circle x ∈ S = R/2πZ. For a dissipative nonlinearity f this equation generates a dissipative semiflow in the appropriate function space, and the corresponding global attractor Af is called a Sturm attractor. If f = f(u, p) is even in p, then the semiflow possesses an embedded flow satisfying Neumann bound...

We have formalized the Sturm-Tarski theorem (also referred as the Tarski theorem): ∑ x∈(a,b),P (x)=0 sign(Q(x)) = Var(SRemS(P, P ′Q; a, b)) where a < b are elements of R ∪ {−∞,∞} that are not roots of P , with P,Q ∈ R[x]. Note, the usual Sturm theorem is an instance of the Sturm-Tarski theorem with Q = 1. The proof is based on [1] and Cyril Cohen’s work in Coq [2]. With the Sturm-Tarki theorem,...

In this paper, we study the inverse problem for Sturm Liouville with conformable fractional differential operators of order and finite number interior discontinuous conditions. For aim first, asymptotic formulas solutions, eigenvalues eigenfunctions are calculated. Then some uniqueness theorems proposed eigenvalue proved. Finally, Hald's theorem &#x0D; Sturm-Liouville is developed.

Consider the minimal Sturm-Liouville operator A = Amin generated by the differential expression

in this paper, we investigate infinite product representation of the solution of a sturm-liouville equation with an indefinite weight function which has two zeros and/or singularitiesin a finite interval. first, by using of the asymptotic estimates provided in [w. eberhard, g.freiling, k. wilcken-stoeber, indefinite eigenvalue problems with several singular pointsand turning points, math. nachr...