1. Sturm-Liouville Operators with Discontinuous Boundary Conditions Aiping Wang 2014.05.09

2. Outline 1. Introduction 2. Background 3. Problem Formulation 4. Main ideas to deal with this problem 5. Main Results

3. 1. Introduction 1. 1 The Regular Sturm-Liouville Case : A regular two point SLP consisits of the equation onwhere ● The general Sturm-Liouville problems

4. Here denotes the Matrices with complex entries. Here we specialize to the self-adjoint case. (A self-adjoint SLP we mean a problem which generates a self-adjoint operator In some Hilbert space.) together with general, not necessarily self-adjoint, two point boundary conditions

5. The Regular Self-Adjoint SLP: Consider the symmetric SL equation

8. All self-Adjoint realizations S of equation (1.1) are characterized by the boundary conditions:

9. Canonical forms of the Regular Self-Adjoint SLP: Separated conditions:

10. Coupled BC:

11. 1.2 The Singlar Sturm-Liouville Case :

12. Singular LC self-adjoint SLP (d=2): Assume that the endpoints a and b are LC.

13. The function u, v can be any

14. Singular LP self-adjoint SLP (d=1): For example: endpoint a is regular, and b is LP:

15. In the classical SL theory, the solutions and their (quasi) derivative are continuous at all interior points on the interval. But these conditions can not be satisfied in many practical problems.

16. The problem we investigated We study regular Sturm-Liouville problems (SLP’s) which have discontinuities at an interior point c. Some conditions are imposed on the interior point c and such conditions involve left and right limits of solutions and their quasi- derivatives at c.

19. 2. Background Physical background

20. Example

25. 3. Problem Formulation

30. We will investigate the

31. 4. Main ideas to deal with this problem

33. 5. Main Results

41. Thanks!