Sturm-Liouville Operators with Discontinuous Boundary Conditions

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Sturm-Liouville Operators withDiscontinuous
Boundary Conditions

Aiping Wang
2014.05.09
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Outline

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

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1. Introduction
? The general Sturm-Liouville problems

• 1. 1 The Regular Sturm-Liouville Case

A regular two point SLP consisits of the equation
on
where
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together with general, not necessarily
self-adjoint, two point boundary conditions
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.)
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• The Regular Self-Adjoint SLP

Consider the symmetric SL equation
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• All self-Adjoint realizations S of equation
  (1.1) are characterized by the boundary
  conditions

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• Canonical forms of the Regular Self-Adjoint SLP
• Separated conditions

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• Coupled BC

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• 1.2 The Singlar Sturm-Liouville Case

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• Singular LC self-adjoint SLP (d2)
• Assume that the endpoints a and b are LC.

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The function u, v can be any
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• Singular LP self-adjoint SLP (d1)

For example endpoint a is regular, and b is LP
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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.
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• The problem we investigated
• We study regular Sturm-Liouville problems
  (SLPs) 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.

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2. Background

• Physical background

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• Example

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3. Problem Formulation
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We will investigate the
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4. Main ideas to deal with this problem
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5. Main Results
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• Thanks!