The Sturm–Liouville hierarchy of evolution equations was introduced in [Adv. Nonlinear Stud. 11 (2011), 555–591] and includes the Korteweg–de Vries and the Camassa– Holm hierarchies. We discuss some solutions of this hierarchy which are obtained as limits of algebro-geometric solutions. The initial data of our solutions are (generalized) reflectionless Sturm–Liouville potentials [Stoch. Dyn. 8 ...

Sturm-Liouville Theory Christopher J. Adkins Master of Science Graduate Department of Mathematics University of Toronto 2014 A basic introduction into Sturm-Liouville Theory. We mostly deal with the general 2ndorder ODE in self-adjoint form. There are a number of things covered including: basic asymptotics, properties of the spectrum, interlacing of zeros, transformation arguments...

In this article, we consider dissipative Sturm–Liouville operators in the limit-circle case on time scales. Then, using the Livšic’s Theorem, we prove the completeness of the system of root vectors for dissipative Sturm–Liouville operators. 2013 Elsevier Inc. All rights reserved.

We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigen-parameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space.

For a system of coupled Schrödinger equations, the relationship is found between the vector-valued norming constants and M + 1 spectra of the M-channel Sturm-Liouville operator with the same potential matrix but different boundary conditions. Under a special choice of particular boundary conditions, this equation for norming vectors has a unique solution. The norming vectors derived from M + 1 ...

By constructing an available integral operator and combining Krasnosel’skii-Zabreiko fixed point theorem with properties of Green’s function, this paper shows the existence of multiple positive solutions for a class of m-point second-order Sturm-Liouville-like boundary value problems on time scales with polynomial nonlinearity. The results significantly extend and improve many known results for...

We consider the problem of how the least eigenvalue of a SturmLiouville problem changes as the coefficients are varied. For a certain class of such problems, it is proved that the least eigenvalue can be maximized by placing constraints on the coefficients. Applications are made to various types of column problems. As a preliminary, we develop the spectral theory for a one term operator which h...

We study the complexity of approximating the smallest eigenvalue of a univariate Sturm-Liouville problem on a quantum computer. This general problem includes the special case of solving a one-dimensional Schrödinger equation with a given potential for the ground state energy. The Sturm-Liouville problem depends on a function q, which, in the case of the Schrödinger equation, can be identified w...

We study Sturm-Liouville operators on closed sets of a special structure, which are sometimes referred as time scales and often appear in modelling various real processes. Depending the set such unify both differential difference operators. obtain properties their spectral characteristics inverse problems with respect to them. prove that uniquely determine operator.

An inverse spectral problem for the Sturm-Liouville operator with a singular potential from class $W_2^{-1}$ is solved by method of mappings. We prove uniqueness theorem, develop constructive algorithm solution and obtain necessary sufficient conditions solvability in self-adjoint non-self-adjoint cases.