DEigensystem
✖
DEigensystem
gives the n smallest magnitude eigenvalues and eigenfunctions for the linear differential operator ℒ over the region Ω.
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 ℒ satisfy ℒ[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
open allclose allBasic Examples (2)Summary of the most common use cases
Find the 4 smallest eigenvalues and eigenfunctions of the Laplacian operator on [0,π]:

https://wolfram.com/xid/0btnz7gc2iv2-yzdn77


https://wolfram.com/xid/0btnz7gc2iv2-wx5bbs

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

https://wolfram.com/xid/0btnz7gc2iv2-d3ex

https://wolfram.com/xid/0btnz7gc2iv2-crtyb1


https://wolfram.com/xid/0btnz7gc2iv2-q0978

Scope (20)Survey of the scope of standard use cases
1D (9)

https://wolfram.com/xid/0btnz7gc2iv2-dd5aze
Specify homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-yjqx5u
Find the 5 smallest eigenvalues and eigenfunctions in an interval:

https://wolfram.com/xid/0btnz7gc2iv2-b0b4f6

https://wolfram.com/xid/0btnz7gc2iv2-cvvh62


https://wolfram.com/xid/0btnz7gc2iv2-ge4exl


https://wolfram.com/xid/0btnz7gc2iv2-2d8bfg
Specify homogeneous Neumann boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-kocy8b
Find the 5 smallest eigenvalues and eigenfunctions in an interval:

https://wolfram.com/xid/0btnz7gc2iv2-v0oeps

https://wolfram.com/xid/0btnz7gc2iv2-3z82ab


https://wolfram.com/xid/0btnz7gc2iv2-ew1oj0

https://wolfram.com/xid/0btnz7gc2iv2-snvhso


https://wolfram.com/xid/0btnz7gc2iv2-cuzc9o
Specify a homogeneous Dirichlet boundary condition:

https://wolfram.com/xid/0btnz7gc2iv2-c9q6xh
Specify a homogeneous Neumann boundary condition:

https://wolfram.com/xid/0btnz7gc2iv2-bg0i4l
Find the 5 smallest eigenvalues and eigenfunctions in an interval:

https://wolfram.com/xid/0btnz7gc2iv2-c5axvx

https://wolfram.com/xid/0btnz7gc2iv2-jy4ton


https://wolfram.com/xid/0btnz7gc2iv2-eb28a


https://wolfram.com/xid/0btnz7gc2iv2-u46ab
Specify a homogeneous Dirichlet boundary condition:

https://wolfram.com/xid/0btnz7gc2iv2-e05awx
Specify a homogeneous non-zero Neumann boundary condition:

https://wolfram.com/xid/0btnz7gc2iv2-hm0vt3
Find the 5 smallest eigenvalues and eigenfunctions in an interval:

https://wolfram.com/xid/0btnz7gc2iv2-lya86
The eigenvalues are roots of a transcendental equation:

https://wolfram.com/xid/0btnz7gc2iv2-nsdqid


https://wolfram.com/xid/0btnz7gc2iv2-id4r85


https://wolfram.com/xid/0btnz7gc2iv2-cldpj3
Specify a homogeneous Dirichlet boundary condition:

https://wolfram.com/xid/0btnz7gc2iv2-essb5
Find the 5 smallest eigenvalues and eigenfunctions in an interval:

https://wolfram.com/xid/0btnz7gc2iv2-by1ifr
The eigenvalues are roots of a transcendental equation:

https://wolfram.com/xid/0btnz7gc2iv2-cnfmi7


https://wolfram.com/xid/0btnz7gc2iv2-iqi360


https://wolfram.com/xid/0btnz7gc2iv2-lzzgzu
Specify a homogeneous Neumann boundary condition:

https://wolfram.com/xid/0btnz7gc2iv2-bzyyfz
Find the 5 smallest eigenvalues and eigenfunctions in an interval:

https://wolfram.com/xid/0btnz7gc2iv2-di935k
The eigenvalues are roots of a transcendental equation:

https://wolfram.com/xid/0btnz7gc2iv2-d60btu


https://wolfram.com/xid/0btnz7gc2iv2-jy8mb2

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

https://wolfram.com/xid/0btnz7gc2iv2-cdj4k3

https://wolfram.com/xid/0btnz7gc2iv2-gdd6d0


https://wolfram.com/xid/0btnz7gc2iv2-j8g1rh

Enter the quantum harmonic operator:

https://wolfram.com/xid/0btnz7gc2iv2-33s8q5
Find symbolic expressions for the eigenvalues and eigenfunctions on the real line:

https://wolfram.com/xid/0btnz7gc2iv2-gjib28

Specify a heat equation with homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-mzmdbz
Find the 4 smallest eigenvalues and eigenfunctions:

https://wolfram.com/xid/0btnz7gc2iv2-1dfazx


https://wolfram.com/xid/0btnz7gc2iv2-ribue

2D (6)
Specify a Laplacian operator with homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-olvz0
Find the 9 smallest eigenvalues and eigenfunctions in a rectangle:

https://wolfram.com/xid/0btnz7gc2iv2-q0svtk

https://wolfram.com/xid/0btnz7gc2iv2-yhxj38


https://wolfram.com/xid/0btnz7gc2iv2-ey9fmu

Specify a Laplacian operator with homogeneous Neumann boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-dzc5oc
Find the 4 smallest eigenvalues and eigenfunctions in a rectangle:

https://wolfram.com/xid/0btnz7gc2iv2-3cf8zi

https://wolfram.com/xid/0btnz7gc2iv2-qpg6gs


https://wolfram.com/xid/0btnz7gc2iv2-snrji


https://wolfram.com/xid/0btnz7gc2iv2-hz8azd


https://wolfram.com/xid/0btnz7gc2iv2-sq6djy
Specify homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-tzp4z
Find the 4 smallest eigenvalues and eigenfunctions of the operator in a unit disk:

https://wolfram.com/xid/0btnz7gc2iv2-646c3s

https://wolfram.com/xid/0btnz7gc2iv2-4ir027


https://wolfram.com/xid/0btnz7gc2iv2-dvcswe

Specify a quantum harmonic oscillator operator:

https://wolfram.com/xid/0btnz7gc2iv2-x4itsm
Specify homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-mdxmrr
Find the 6 smallest eigenvalues and eigenfunctions of the operator on the plane:

https://wolfram.com/xid/0btnz7gc2iv2-tdyb7u

https://wolfram.com/xid/0btnz7gc2iv2-vmgimx


https://wolfram.com/xid/0btnz7gc2iv2-wiu9y7


https://wolfram.com/xid/0btnz7gc2iv2-calu3w
Specify homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-bs5esh
Find the 6 smallest eigenvalues and eigenfunctions of the operator in a triangle:

https://wolfram.com/xid/0btnz7gc2iv2-bwxx0l

https://wolfram.com/xid/0btnz7gc2iv2-l49dkt


https://wolfram.com/xid/0btnz7gc2iv2-bp7icj


https://wolfram.com/xid/0btnz7gc2iv2-hqm92
Specify homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-kd8b62
Find the 4 smallest eigenvalues and eigenfunctions of the operator in a sector of a disk:

https://wolfram.com/xid/0btnz7gc2iv2-fc0694

https://wolfram.com/xid/0btnz7gc2iv2-b0jk9w


https://wolfram.com/xid/0btnz7gc2iv2-lyqv9v

3D (5)

https://wolfram.com/xid/0btnz7gc2iv2-5soch
Specify homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-idhpb
Find the 7 smallest eigenvalues and eigenfunctions in a cuboid:

https://wolfram.com/xid/0btnz7gc2iv2-lqu653

https://wolfram.com/xid/0btnz7gc2iv2-beoef7


https://wolfram.com/xid/0btnz7gc2iv2-coor33


https://wolfram.com/xid/0btnz7gc2iv2-d0ilyf
Specify homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-i9ulj9
Find the 7 smallest eigenvalues and eigenfunctions in a cylinder:

https://wolfram.com/xid/0btnz7gc2iv2-jo5954

https://wolfram.com/xid/0btnz7gc2iv2-c137sl


https://wolfram.com/xid/0btnz7gc2iv2-4wlns


https://wolfram.com/xid/0btnz7gc2iv2-chbrb7
Specify homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-cwisu0
Find the 7 smallest eigenvalues and eigenfunctions in a ball:

https://wolfram.com/xid/0btnz7gc2iv2-j6wop0

https://wolfram.com/xid/0btnz7gc2iv2-bhg57v


https://wolfram.com/xid/0btnz7gc2iv2-hhf5pu
Specify homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-tdyqt
Find the 7 smallest eigenvalues and eigenfunctions in a prism:

https://wolfram.com/xid/0btnz7gc2iv2-j4mv75

https://wolfram.com/xid/0btnz7gc2iv2-jct22c


https://wolfram.com/xid/0btnz7gc2iv2-bewipj

Specify a quantum harmonic oscillator operator:

https://wolfram.com/xid/0btnz7gc2iv2-o98g3b
Specify homogeneous Dirichlet boundary conditions:

https://wolfram.com/xid/0btnz7gc2iv2-8oi6tm
Find the 8 smallest eigenvalues and eigenfunctions throughout space:

https://wolfram.com/xid/0btnz7gc2iv2-gp270a

https://wolfram.com/xid/0btnz7gc2iv2-evmcm5


https://wolfram.com/xid/0btnz7gc2iv2-2klld7

Options (2)Common values & functionality for each option
Assumptions (1)
Use Assumptions to simplify a result:

https://wolfram.com/xid/0btnz7gc2iv2-ecanzm

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

https://wolfram.com/xid/0btnz7gc2iv2-yyrzi2

Applications (3)Sample problems that can be solved with this function
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
:

https://wolfram.com/xid/0btnz7gc2iv2-chgfue

Compute the Fourier coefficients:

https://wolfram.com/xid/0btnz7gc2iv2-euk8i9

https://wolfram.com/xid/0btnz7gc2iv2-d8ezwq

The required eigenfunction expansion is:

https://wolfram.com/xid/0btnz7gc2iv2-khjz1n

Compare the function with its eigenfunction expansion:

https://wolfram.com/xid/0btnz7gc2iv2-byveyd

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

https://wolfram.com/xid/0btnz7gc2iv2-ee0l91

https://wolfram.com/xid/0btnz7gc2iv2-qsijm

Form a linear combination of eigenfunctions:

https://wolfram.com/xid/0btnz7gc2iv2-q3w8c

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

https://wolfram.com/xid/0btnz7gc2iv2-h2j74j

The solution satisfies the homogeneous Dirichlet condition:

https://wolfram.com/xid/0btnz7gc2iv2-bjjefv


https://wolfram.com/xid/0btnz7gc2iv2-d3v70g

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:

https://wolfram.com/xid/0btnz7gc2iv2-551rrk
Compute the eigenvalues—the energies of the respective states—and normalized eigenfunctions:

https://wolfram.com/xid/0btnz7gc2iv2-jxv26o

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

https://wolfram.com/xid/0btnz7gc2iv2-qvnvu

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

https://wolfram.com/xid/0btnz7gc2iv2-oz2boj


https://wolfram.com/xid/0btnz7gc2iv2-nbi4dx


https://wolfram.com/xid/0btnz7gc2iv2-9qfa20

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

https://wolfram.com/xid/0btnz7gc2iv2-rxoenq

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

https://wolfram.com/xid/0btnz7gc2iv2-mwva1k

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

https://wolfram.com/xid/0btnz7gc2iv2-cqpldr

Visualize the probability density over time:

https://wolfram.com/xid/0btnz7gc2iv2-s7tsiv

Properties & Relations (6)Properties of the function, and connections to other functions
Use NDEigensystem to find numerical eigenvalues and eigenvectors:

https://wolfram.com/xid/0btnz7gc2iv2-b2g6ab
Find exact eigenvalues and eigenvectors:

https://wolfram.com/xid/0btnz7gc2iv2-h80wjy


https://wolfram.com/xid/0btnz7gc2iv2-bnu8sm

Find numerical eigenvalues and eigenvectors:

https://wolfram.com/xid/0btnz7gc2iv2-fub3ug

Use DEigenvalues to find the eigenvalues for a differential operator:

https://wolfram.com/xid/0btnz7gc2iv2-ls3dpn
Find eigenvalues and eigenfunctions:

https://wolfram.com/xid/0btnz7gc2iv2-efw01u


https://wolfram.com/xid/0btnz7gc2iv2-gdvfkv


https://wolfram.com/xid/0btnz7gc2iv2-cb55dh

Use DSolve to solve an eigenvalue problem:

https://wolfram.com/xid/0btnz7gc2iv2-hoe9ag
Find eigenvalues and eigenfunctions:

https://wolfram.com/xid/0btnz7gc2iv2-bqrzjm

Find the complete eigensystem:

https://wolfram.com/xid/0btnz7gc2iv2-es16mg

The eigenfunctions given by DEigensystem are orthogonal:

https://wolfram.com/xid/0btnz7gc2iv2-bbit9o
Find eigenvalues and eigenfunctions:

https://wolfram.com/xid/0btnz7gc2iv2-i15m9d

Verify that the eigenfunctions are orthogonal:

https://wolfram.com/xid/0btnz7gc2iv2-dfszts

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

https://wolfram.com/xid/0btnz7gc2iv2-gx3hrl
Find eigenvalues and eigenfunctions:

https://wolfram.com/xid/0btnz7gc2iv2-uv309

The eigenfunctions are not orthonormalized by default:

https://wolfram.com/xid/0btnz7gc2iv2-kwuivh

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

https://wolfram.com/xid/0btnz7gc2iv2-gzxmv


https://wolfram.com/xid/0btnz7gc2iv2-htdmtj

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

https://wolfram.com/xid/0btnz7gc2iv2-ktfv0


https://wolfram.com/xid/0btnz7gc2iv2-logptw

Possible Issues (2)Common pitfalls and unexpected behavior
Inhomogeneous Dirichlet conditions are replaced with homogeneous ones:

https://wolfram.com/xid/0btnz7gc2iv2-d7hkcu



https://wolfram.com/xid/0btnz7gc2iv2-jg3nz5

Inhomogeneous Neumann values are replaced with homogeneous ones:

https://wolfram.com/xid/0btnz7gc2iv2-vkj213



https://wolfram.com/xid/0btnz7gc2iv2-0cm5i9

Wolfram Research (2015), DEigensystem, Wolfram Language function, https://reference.wolfram.com/language/ref/DEigensystem.html.
Text
Wolfram Research (2015), DEigensystem, Wolfram Language function, https://reference.wolfram.com/language/ref/DEigensystem.html.
Wolfram Research (2015), DEigensystem, Wolfram Language function, https://reference.wolfram.com/language/ref/DEigensystem.html.
CMS
Wolfram Language. 2015. "DEigensystem." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DEigensystem.html.
Wolfram Language. 2015. "DEigensystem." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DEigensystem.html.
APA
Wolfram Language. (2015). DEigensystem. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DEigensystem.html
Wolfram Language. (2015). DEigensystem. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DEigensystem.html
BibTeX
@misc{reference.wolfram_2025_deigensystem, author="Wolfram Research", title="{DEigensystem}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/DEigensystem.html}", note=[Accessed: 08-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_deigensystem, organization={Wolfram Research}, title={DEigensystem}, year={2015}, url={https://reference.wolfram.com/language/ref/DEigensystem.html}, note=[Accessed: 08-June-2025
]}