How do you solve the Sturm-Liouville problem?
These equations give a regular Sturm-Liouville problem. Identify p,q,r,αj,βj in the example above. y(x)=Acos(√λx)+Bsin(√λx)if λ>0,y(x)=Ax+Bif λ=0. Let us see if λ=0 is an eigenvalue: We must satisfy 0=hB−A and A=0, hence B=0 (as h>0), therefore, 0 is not an eigenvalue (no nonzero solution, so no eigenfunction).
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.
What is an eigenvalue of the Sturm-Liouville problem?
The eigenvalues of a Sturm-Liouville problem are the values of λ for which nonzero solutions exist.
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.
Is Liouville operator Hermitian?
L is a Hermitian operator. Hence all its eigenvalues are real. Hence ρ(X,t) cannot relax to equilibrium in any obvious way. The Liouville equation reflects the time reversal symmetry of the underlying microscopic dynamics.
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.
What is boundary value problem in differential equations?
A Boundary value problem is a system of ordinary differential equations with solution and derivative values specified at more than one point. Most commonly, the solution and derivatives are specified at just two points (the boundaries) defining a two-point boundary value problem.
What do you mean by Dirichlet problem in spherical regions?
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. This requirement is called the Dirichlet boundary condition.
What is Liouville equation?
The Liouville equation is a partial differential equation for the phase space probability distribution function. Thus, it specifies a general class of functions f(x,t) that satisfy it.
What is density matrix in quantum mechanics?
The density matrix or density operator is an alternate representation of the state of a quantum system for which we have previously used the wavefunction. The density matrix is formally defined as the outer product of the wavefunction and its conjugate. ρ(t)≡ ψ (t) ψ (t) .
Why is boundary value a problem?
A Sturm-Liouville problem consists of A Sturm-Liouville equation on an interval: (p(x)y′)′+(q(x) +λr(x))y = 0, a < x < b, (1) together with Boundary conditions, i.e. specified behavior of y at x = a and x = b. We will assume that p, p′, q and r are continuous and p > 0 on (at least) the open interval a < x < b.
What is an example of a Sturm-Liouville equation?
Because λ is a parameter, it is frequently replaced by other variables or expressions. Many “familiar” ODEs that occur during separation of variables can be put in Sturm-Liouville form. Example Show that y′′+λy = 0 is a Sturm-Liouville equation. We simply take p(x) = r(x) = 1 and q(x) = 0. Daileda Sturm-Liouville Theory
What are Chebyshev’s polynomials?
Chebyshev Polynomials: For , , , , , and , the Sturm-Liouville equation becomes the Chebyshev’s differential equation which is defined on . The solutions of the Chebyshev’s differential equation with is called Chebyshev Polynomials which form a complete orthogonal set on the interval with respect to .