What is Sturm Liouville problem explain?

Sturm-Liouville problem, or eigenvalue problem, in mathematics, a certain class of partial differential equations (PDEs) subject to extra constraints, known as boundary values, on the solutions.

Why is Sturm Liouville important?

The methods and notions that originated during studies of the Sturm–Liouville problem played an important role in the development of many directions in mathematics and physics. It was and remains a constant source of new ideas and problems in the spectral theory of operators and in related problems in analysis.

What is Sturm Liouville differential equation?

In mathematics and its applications, classical Sturm–Liouville theory is the theory of real second-order linear ordinary differential equations of the form: (1) for given coefficient functions p(x), q(x), and w(x) and an unknown function y of the free variable x.

What is Sturm-Liouville eigenvalue problem?

The problem of finding a complex number µ if any, such that the BVP (6.2)-(6.3) with λ = µ, has a non-trivial solution is called a Sturm-Liouville Eigen Value Problem (SL-EVP). Such a value µ is called an eigenvalue and the corresponding non-trivial solutions y(.; µ) are called eigenfunctions.

What are the eigenvalues and eigenfunctions of the Sturm-Liouville problem?

(p(x)y′)′ + (q(x) + λr(x))y = 0, a < x < b, (plus boundary conditions), is called an eigenfunction, and the corresponding value of λ is called its eigenvalue. The eigenvalues of a Sturm-Liouville problem are the values of λ for which nonzero solutions exist.

Is the Schrodinger equation Sturm-Liouville?

In fact, a Schrödinger equation in the coordinate representation can be seen as a Sturm-Liouville differential equation. It means that there is an Sturm-Liouville (SL) operator (a differential operator) which obeys an eigenvalue equation.

Is the Sturm-Liouville operator Hermitian?

3 Hermitian Sturm Liouville operators. In mathematical physics the domain is often delimited by points a and b where p(a)=p(b)=0. If we then add a boundary condition that w(x)p(x) and w (x)p(x) are finite (or a specific finite number) as x→a b for all solutions w(x), the operator is Hermitian.

What is eigenfunction and eigenvalue?

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.

What is eigenvalue Schrodinger equation?

Both time-dependent and time-independent Schrödinger equations are the best known instances of an eigenvalue equations in quantum mechanics, with its eigenvalues corresponding to the allowed energy levels of the quantum system. The object on the left that acts on ψ(x) is an example of an operator.

What is the meaning of eigenfunction?

Definition of eigenfunction : the solution of a differential equation (such as the Schrödinger wave equation) satisfying specified conditions.

What is eigenfunction in Schrodinger wave equation?

2. If a function does, then ψ is known as an eigenfunction and the constant k is called its eigenvalue (these terms are hybrids with German, the purely English equivalents being “characteristic function” and “characteristic value”, respectively). Solving eigenvalue problems are discussed in most linear algebra courses.

What is the classical Sturm-Liouville theory?

In mathematics and its applications, classical Sturm–Liouville theory is the theory of real second-order linear ordinary differential equations of the form: for given coefficient functions p(x), q(x), and w(x) and an unknown function y of the free variable x. The function w(x), sometimes denoted r(x), is called the weight or density function.

What is the Sturm-Liouville form?

Orthogonality Sturm-Liouville problems Eigenvalues and eigenfunctions. Sturm-Liouville equations. A Sturm-Liouville equation is a second order linear differential equation that can be written in the form (p(x)y′)′ +(q(x) +λr(x))y = 0. Such an equation is said to be in Sturm-Liouville form.

How do you find the multiplicity of a Sturm Liouville problem?

In Sturm-Liouville theory, we say that the multiplicity of an eigenvalue of a Sturm-Liouville problem L[˚] = r(x)˚(x) a 1˚(0) + a 2˚0(0) = 0 b 1˚(1) + b 2˚0(1) = 0 if there are exactly mlinearly independent solutions for that value of . Theorem 12.7. The eigenvalues of a Sturm-Liouville problem are all of multiplicity one. Moreover, the

How do you prove that the eigenvalues of a Sturm–Liouville operator are real?

This can be seen formally by using integration by parts twice, where the boundary terms vanish by virtue of the boundary conditions. It then follows that the eigenvalues of a Sturm–Liouville operator are real and that eigenfunctions of L corresponding to different eigenvalues are orthogonal.