# Regular Sturm-Liouville, one-dimensional eigenfunctspace

• usn7564
In summary, the Sturm-Liouville problem is a boundary value problem where a differential equation involving r(x), q(x), and p(x) is satisfied over an interval [a, b]. The additional condition that r(a)>0 (or r(b)>0) guarantees that the problem cannot have two linearly independent eigenfunctions corresponding to the same eigenvalue. This is proven by showing that if two eigenfunctions, X and Y, have the same eigenvalue, then the function Z = Y(a)X(x) - X(a)Y(x) also satisfies the problem with boundary conditions z(a) and z'(a)=0. By uniqueness, it follows that z(x) must be identically zero over the interval [a
usn7564
Sturm-Liouville problem:
$$\frac{d}{dx} (r(x) \frac{dX}{dx}) + q(x)X(x) + \lambda p(x) X(x) = 0 \quad x \in \left[a, b \right]$$
$$a_1 X(a) + a_2X'(a) = 0$$
$$b_1 X(b) + b_2X'(b) = 0$$
$$r, r', q, p \in \mathbb{C} \forall x \in in \left[a, b \right]$$

Theorem:
Under the additional condition that r(a)>0 (or r(b) > 0), the S.L BVP cannot have two linearly independent eigenfunctions corresponding to the same eigenvalue.

Proof:
X, Y two eigenfunctions corresponding to the same eigenvalue. Then Z = Y(a)X(x) - X(a)Y(x) is also a solution and it can be shown that [skipping how to get z(a), z'(a) as there's no real problem there for me]

$$z(a) = z'(a) = 0 \implies z(x) = 0 \forall x \in \left[a, b \right]$$
QED, catfoots and other scribbles.

Question:
I don't get the final implication at all. Why does knowing that z(a) and z'(a) = 0 at x = a show that z(x) is identically zero over the interval in question?

Last edited:
z is a solution to the SL problem with boundary conditions z(a) and z'(a)=0. The trivial solution f(x)=0 also is. Uniqueness leads to z=0.

usn7564
Explains why it wasn't elaborated further, feel silly now. Thank you, need to remind myself of BVP's it seems.

## 1. What is a Sturm-Liouville equation?

A Sturm-Liouville equation is a type of differential equation that is used in physics and mathematics to describe the behavior of oscillating systems. It takes the form of a second-order linear differential equation and has specific boundary conditions that must be satisfied.

## 2. What is a regular Sturm-Liouville problem?

A regular Sturm-Liouville problem is a type of Sturm-Liouville equation in which the differential equation and boundary conditions are well-behaved and have a unique solution. This means that the eigenfunctions (solutions to the equation) form a complete set and can be used to represent any arbitrary function within the problem's domain.

## 3. What is the one-dimensional eigenfunction space in regular Sturm-Liouville problems?

The one-dimensional eigenfunction space in regular Sturm-Liouville problems refers to the set of eigenfunctions that satisfy the given Sturm-Liouville equation and boundary conditions. These eigenfunctions form a basis for the solution space and can be used to represent any function within the problem's domain.

## 4. How are eigenfunctions and eigenvalues related in Sturm-Liouville problems?

In Sturm-Liouville problems, eigenfunctions (solutions to the differential equation) are associated with corresponding eigenvalues. These eigenvalues represent the different frequencies at which the system can oscillate, and the eigenfunctions determine the shape or form of the oscillations at each frequency.

## 5. What are some applications of regular Sturm-Liouville problems?

Regular Sturm-Liouville problems have many applications in physics and engineering, including the analysis of vibrating systems, heat transfer, and quantum mechanics. They are also used in mathematical fields such as Fourier analysis and spectral theory to study the properties of differential operators.

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