Purchase Solution

Eigenfunctons and eigenvalues of d^2/dx^2

Not what you're looking for?

Ask Custom Question

Consider the square of the derivative operator D^2

(a) Show that D^2 is a linear operator
(b) Find the eigenfunctions and corresponding eigenvalues of D^2.
(c) Give an example of an eigenfunction of D^2 which is not an eigenfunction of D.

Purchase this Solution
Solution Summary

A detailed solution is given. The expert considers the square of the derivative operators and linear operators.

Solution Preview

D is a linear operator. If f and g are two functions and a and b two complex numbers, then:

D[a f + b g] = a D[f] + b D[g] (1)

Which is the defining property of a linear operator (or equivalent to other set of defining properties)

The operator D^2 is defined as:

D^2f = ...

Purchase this Solution
Free BrainMass Quizzes
Basic Physics

This quiz will test your knowledge about basic Physics.

The Moon

Test your knowledge of moon phases and movement.

Introduction to Nanotechnology/Nanomaterials

This quiz is for any area of science. Test yourself to see what knowledge of nanotechnology you have. This content will also make you familiar with basic concepts of nanotechnology.

Classical Mechanics

This quiz is designed to test and improve your knowledge on Classical Mechanics.

Intro to the Physics Waves

Some short-answer questions involving the basic vocabulary of string, sound, and water waves.

View More Free Quizzes
Related BrainMass Solutions

  • Derivatives and Matrices

    518480 Derivatives and Matrices f(x,y) = 3/x + 5xy +2/y The 2nd derivative of f at this point is D^2f(x*,y*) = A 2 x 2 matrix I know the top right and bottom right are 5, but I am missing the top and bottom left.

  • Green Functions

    lims 0 to t lims 0 to t Find the GF sol when f(x)=sinh t Find the eigenvalues and normalised eigenfunctions of d(^2)y/dx(^2) + 4*y =Lambda y then use this result to write the GF of d(^2)y/dx(^2) + 4*y=f(x) and say why it is appropriate

  • Conic graph

    Do this by writing this equation in matrix form; then change the equation to a sum of squares of the form x'^T Dx' where D is a diagonal matrix. Solution.

  • 2x2 Symmetric Matrix with Positive Eigenvalues.

    517973 Symmetric Matrix with Positive Eigenvalues Suppose that both eigenvalues of a 2 x 2 symmetric matrix B are positive. Which of the following statements are true? i) The system Bx = b has a unique solution for each b.

  • Differential Equations: Populations

    How would I determine all the equilibrium points for this system of differential equations expressing my answer in terms of p, q, r and s. How would I classify each equilibrium point using the method of matrices with two real distinct eigenvalues?

  • Differential Operators : Eigenvalues and Eigenfunctions

    (a) For the above differential operator FIND S* for the adjoint with respect to (v,u) =S 1-->0 v-bar u dx and compare S with S*. (b) COMPARE the eigenvalues of Lu =... with the eigenvalues of L*v n = ...

  • solving system of differential equations and eigenvalues

    =0 must have complex eigenvalues. 3) Let A = (a b; c d), define the trace of A to be tr(A) = a +d. show that A has only one eignevlue if and only if (tr(A))^2 - 4det(A) = 0 Please see the attached file for detailed solution.

  • Ground state energy for particle using variationa method

    To evaluate the kinetic energy we need the second derivative of psi(x). You can simplify the calculation by writing this as: d^psi/dx2 = -d^3 f/(d alpha dx^2) where f(x) = exp(-alpha x) and differentiating f w.r.t. x first.

  • Orthogonal basis and matrix, Gram-Schmidt process

    If you answer false, find a counter example and show the transformation is not linear. 4. Let f = f(x) and g = g(x) be two functions in C[a,b] and define = Integral with limits a to b of f(x)g(x) dx.

View More Related Solutions