# DEigensystem

DEigensystem[[u[x,y,]],u,{x,y,}Ω,n]

gives the n smallest magnitude eigenvalues and eigenfunctions for the linear differential operator over the region Ω.

DEigensystem[eqns,u,t,{x,y,}Ω,n]

gives the eigenvalues and eigenfunctions for solutions u of the time-dependent differential equations eqns.

# Details and Options • DEigensystem can compute eigenvalues and eigenfunctions for ordinary and partial differential operators with given boundary conditions.
• DEigensystem gives lists {{λ1,,λn},{u1,,un}} of eigenvalues λi and eigenfunctions ui.
• An eigenvalue and eigenfunction pair {λi,ui} for the differential operator satisfy [ui[x,y,]]==λi ui[x,y,].
• Homogeneous DirichletCondition or NeumannValue boundary conditions may be included. Inhomogeneous boundary conditions will be replaced with corresponding homogeneous boundary conditions.
• When no boundary condition is specified on the boundary Ω, then this is equivalent to specifying a Neumann 0 condition.
• The equations eqns are specified as in DSolve.
• N[DEigensystem[]] calls NDEigensystem for eigensystems that cannot be computed symbolically.
• The following options can be given:
•  Assumptions \$Assumptions assumptions on parameters Method Automatic what method to use
• Eigenfunctions are not automatically normalized. The setting Method->"Normalize" can be used to give normalized eigenfunctions.

# Examples

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## Basic Examples(2)

Find the 4 smallest eigenvalues and eigenfunctions of the Laplacian operator on [0,π]:

Visualize the eigenfunctions:

Compute the first 6 eigenfunctions for a circular membrane with the edges clamped:

Visualize the eigenfunctions:

## Scope(20)

### 1D(9)

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

Visualize the eigenfunctions:

Specify a Laplacian operator:

Specify homogeneous Neumann boundary conditions:

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

This is equivalent:

Specify a Laplacian operator:

Specify a homogeneous Dirichlet boundary condition:

Specify a homogeneous Neumann boundary condition:

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

Visualize the eigenfunctions:

Specify a Laplacian operator:

Specify a homogeneous Dirichlet boundary condition:

Specify a homogeneous non-zero Neumann boundary condition:

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

The eigenvalues are roots of a transcendental equation:

Visualize the eigenfunctions:

Specify an Airy operator:

Specify a homogeneous Dirichlet boundary condition:

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

The eigenvalues are roots of a transcendental equation:

Visualize the eigenfunctions:

Specify an Airy operator:

Specify a homogeneous Neumann boundary condition:

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

The eigenvalues are roots of a transcendental equation:

Visualize the eigenfunctions:

Find symbolic expressions for the eigenvalues and eigenfunctions of a Laplace operator:

Symbolic eigenvalues:

Symbolic eigenfunctions:

Enter the quantum harmonic operator:

Find symbolic expressions for the eigenvalues and eigenfunctions on the real line:

Specify a heat equation with homogeneous Dirichlet boundary conditions:

Find the 4 smallest eigenvalues and eigenfunctions:

Visualize the eigenfunctions:

### 2D(6)

Specify a Laplacian operator with homogeneous Dirichlet boundary conditions:

Find the 9 smallest eigenvalues and eigenfunctions in a rectangle:

Visualize the eigenfunctions:

Specify a Laplacian operator with homogeneous Neumann boundary conditions:

Find the 4 smallest eigenvalues and eigenfunctions in a rectangle:

Visualize the eigenfunctions:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 4 smallest eigenvalues and eigenfunctions of the operator in a unit disk:

Visualize the eigenfunctions:

Specify a quantum harmonic oscillator operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 6 smallest eigenvalues and eigenfunctions of the operator on the plane:

Visualize the eigenfunctions:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 6 smallest eigenvalues and eigenfunctions of the operator in a triangle:

Visualize the eigenfunctions:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 4 smallest eigenvalues and eigenfunctions of the operator in a sector of a disk:

Visualize the eigenfunctions:

### 3D(5)

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 7 smallest eigenvalues and eigenfunctions in a cuboid:

Visualize an eigenfunction:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 7 smallest eigenvalues and eigenfunctions in a cylinder:

Visualize an eigenfunction:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 7 smallest eigenvalues and eigenfunctions in a ball:

Visualize an eigenfunction:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 7 smallest eigenvalues and eigenfunctions in a prism:

Visualize an eigenfunction:

Specify a quantum harmonic oscillator operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 8 smallest eigenvalues and eigenfunctions throughout space:

Visualize the eigenfunctions:

## Options(2)

### Assumptions(1)

Use Assumptions to simplify a result:

Without the option, an equivalent but more complicated answer would be returned:

### Method(1)

Normalize the eigenfunctions for a differential operator:

Verify that the eigenfunctions are normalized to unity:

## Applications(3)

Compute the first three terms in the eigenfunction expansion of the function with respect to the basis provided by a 1D Laplacian with a Dirichlet condition on the interval :

Compute the Fourier coefficients:

The required eigenfunction expansion is:

Compare the function with its eigenfunction expansion:

Build a solution of the heat equation by using a linear combination of eigenfunctions for the heat equation with a Dirichlet condition:

Form a linear combination of eigenfunctions:

Verify that this is indeed a solution of the heat equation:

The solution satisfies the homogeneous Dirichlet condition:

Visualize the solution:

Experimentally, a CO molecule oscillates about its equilibrium length with an effective spring constant of . The oscillations will be governed by the quantum harmonic oscillator equation. In the following, is the reduced mass of the molecule, is the natural frequency, is the displacement from the equilibrium position, and is the reduced Planck's constant:

Compute the eigenvaluesthe energies of the respective statesand normalized eigenfunctions:

If the particle is in an equal superposition of the four states, the wavefunction has the following form:

Compute , , and using base units of atomic mass units, femtoseconds, and picometers, which give values close to order unity:

The response of the eigenfunctions to the potential energy can be visualized by rescaling them to fit in the band :

The probability density function of the displacement from equilibrium is given by :

As a probability distribution, the integral of over the reals is 1 for all :

Visualize the probability density over time:

## Properties & Relations(6)

Use NDEigensystem to find numerical eigenvalues and eigenvectors:

Find exact eigenvalues and eigenvectors:

Find numerical eigenvalues and eigenvectors:

Use DEigenvalues to find the eigenvalues for a differential operator:

Find eigenvalues and eigenfunctions:

Find eigenvalues only:

Use DSolve to solve an eigenvalue problem:

Find eigenvalues and eigenfunctions:

Find the complete eigensystem:

The eigenfunctions given by DEigensystem are orthogonal:

Find eigenvalues and eigenfunctions:

Verify that the eigenfunctions are orthogonal:

The system of eigenfunctions given by DEigensystem is not orthonormal by default:

Find eigenvalues and eigenfunctions:

The eigenfunctions are not orthonormalized by default:

Use Method->"Normalize" to obtain an orthonormal system:

Apply N[DEigensystem[...]] to invoke NDEigensystem if symbolic evaluation fails:

## Possible Issues(2)

Inhomogeneous Dirichlet conditions are replaced with homogeneous ones: The same result:

Inhomogeneous Neumann values are replaced with homogeneous ones: The same result: