Revisions to Eigenvalue problem

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edited Jun 17, 2019 at 15:37

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my question is about solving an eigenvalue problem of the Helmholtz equation using sinc approximation
$\nabla^2E + V (x) = \lambda E$ and $V(x)= X^2 / 2$

I have a problem in calculating the eigenvalues of this system because the eigen values if this system is known to be $\lambda$=$ \{1/2 , 3/2 , 5/2\}$

my code is :

Nn = 8
   h = \[Pi]π/Sqrt[2 Nn]
   h1 = h

   exactdelta2 = h1^2 D[S1[\[Phi]1[x]D[S1[ϕ1[x], j, h1], { \[Phi]1[x]ϕ1[x], 2}]

   ex5 = exactdelta2 /. x -> \[Psi]1[x]ψ1[x] /. \[Phi]1[\[Psi]1[x_]]ϕ1[ψ1[x_]] :> x /. 
  x -> k h1

   ex6 = ex5 /. S1 -> Function[{x, k, h}, Sinc[\[Pi]Sinc[π (x - k h)/h]]

   nabla = Table[ex6, {k, -Nn, Nn}, {j, -Nn, Nn}] /. 
   Indeterminate :> Limit[ex6, k -> j] 

   vx[x_] = x^2/2 /. x -> k h

   vx1 = Table[vx[x], {k, -Nn, Nn}]
   
   vxmat = DiagonalMatrix[vx1]

   A = nabla + vxmat // N
   
   lamda = Eigenvalues[A]

my question is about solving an eigenvalue problem of the Helmholtz equation using sinc approximation
$\nabla^2E + V (x) = \lambda E$ and $V(x)= X^2 / 2$

I have a problem in calculating the eigenvalues of this system because the eigen values if this system is known to be $\lambda$=$ \{1/2 , 3/2 , 5/2\}$

my code is :

Nn = 8
   h = \[Pi]/Sqrt[2 Nn]
   h1 = h

   exactdelta2 = h1^2 D[S1[\[Phi]1[x], j, h1], { \[Phi]1[x], 2}]

   ex5 = exactdelta2 /. x -> \[Psi]1[x] /. \[Phi]1[\[Psi]1[x_]] :> x /. 
  x -> k h1

   ex6 = ex5 /. S1 -> Function[{x, k, h}, Sinc[\[Pi] (x - k h)/h]]

   nabla = Table[ex6, {k, -Nn, Nn}, {j, -Nn, Nn}] /. 
   Indeterminate :> Limit[ex6, k -> j] 

   vx[x_] = x^2/2 /. x -> k h

   vx1 = Table[vx[x], {k, -Nn, Nn}]
   
   vxmat = DiagonalMatrix[vx1]

   A = nabla + vxmat // N
   
   lamda = Eigenvalues[A]

my question is about solving an eigenvalue problem of the Helmholtz equation using sinc approximation
$\nabla^2E + V (x) = \lambda E$ and $V(x)= X^2 / 2$

I have a problem in calculating the eigenvalues of this system because the eigen values if this system is known to be $\lambda$=$ \{1/2 , 3/2 , 5/2\}$

my code is :

Nn = 8
h = π/Sqrt[2 Nn]
h1 = h

exactdelta2 = h1^2 D[S1[ϕ1[x], j, h1], { ϕ1[x], 2}]

ex5 = exactdelta2 /. x -> ψ1[x] /. ϕ1[ψ1[x_]] :> x /. x -> k h1

ex6 = ex5 /. S1 -> Function[{x, k, h}, Sinc[π (x - k h)/h]]

nabla = Table[ex6, {k, -Nn, Nn}, {j, -Nn, Nn}] /.Indeterminate :> Limit[ex6, k -> j] 

vx[x_] = x^2/2 /. x -> k h

vx1 = Table[vx[x], {k, -Nn, Nn}]
   
vxmat = DiagonalMatrix[vx1]

A = nabla + vxmat // N
   
lamda = Eigenvalues[A]

corrected math expression(mathjax)

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edit approved Jun 16, 2019 at 13:29

Xminer

1.2k
8
15

my question is about solving an eigenvalue problem of the Helmholtz equation using sinc approximation [Del]^2E + V (x) = [Lambda] E and V(x)= X^2sinc approximation / 2
$\nabla^2E + V (x) = \lambda E$ and $V(x)= X^2 / 2$

I have a problem in calculating the eigenvalues of this system because the eigen values if this system is known to be [Lambda]= {1/2 , 3/2 , 5/2}$\lambda$=$ \{1/2 , 3/2 , 5/2\}$

my code is :

Nn = 8
   h = \[Pi]/Sqrt[2 Nn]
   h1 = h

   exactdelta2 = h1^2 D[S1[\[Phi]1[x], j, h1], { \[Phi]1[x], 2}]

   ex5 = exactdelta2 /. x -> \[Psi]1[x] /. \[Phi]1[\[Psi]1[x_]] :> x /. 
  x -> k h1

   ex6 = ex5 /. S1 -> Function[{x, k, h}, Sinc[\[Pi] (x - k h)/h]]

   nabla = Table[ex6, {k, -Nn, Nn}, {j, -Nn, Nn}] /. 
   Indeterminate :> Limit[ex6, k -> j] 

   vx[x_] = x^2/2 /. x -> k h

   vx1 = Table[vx[x], {k, -Nn, Nn}]
   
   vxmat = DiagonalMatrix[vx1]

   A = nabla + vxmat // N
   
   lamda = Eigenvalues[A]

my question is about solving an eigenvalue problem of the Helmholtz equation using sinc approximation [Del]^2E + V (x) = [Lambda] E and V(x)= X^2 / 2

I have a problem in calculating the eigenvalues of this system because the eigen values if this system is known to be [Lambda]= {1/2 , 3/2 , 5/2}

my code is :

Nn = 8
   h = \[Pi]/Sqrt[2 Nn]
   h1 = h

   exactdelta2 = h1^2 D[S1[\[Phi]1[x], j, h1], { \[Phi]1[x], 2}]

   ex5 = exactdelta2 /. x -> \[Psi]1[x] /. \[Phi]1[\[Psi]1[x_]] :> x /. 
  x -> k h1

   ex6 = ex5 /. S1 -> Function[{x, k, h}, Sinc[\[Pi] (x - k h)/h]]

   nabla = Table[ex6, {k, -Nn, Nn}, {j, -Nn, Nn}] /. 
   Indeterminate :> Limit[ex6, k -> j] 

   vx[x_] = x^2/2 /. x -> k h

   vx1 = Table[vx[x], {k, -Nn, Nn}]
   
   vxmat = DiagonalMatrix[vx1]

   A = nabla + vxmat // N
   
   lamda = Eigenvalues[A]

my question is about solving an eigenvalue problem of the Helmholtz equation using sinc approximation
$\nabla^2E + V (x) = \lambda E$ and $V(x)= X^2 / 2$

I have a problem in calculating the eigenvalues of this system because the eigen values if this system is known to be $\lambda$=$ \{1/2 , 3/2 , 5/2\}$

my code is :

Nn = 8
   h = \[Pi]/Sqrt[2 Nn]
   h1 = h

   exactdelta2 = h1^2 D[S1[\[Phi]1[x], j, h1], { \[Phi]1[x], 2}]

   ex5 = exactdelta2 /. x -> \[Psi]1[x] /. \[Phi]1[\[Psi]1[x_]] :> x /. 
  x -> k h1

   ex6 = ex5 /. S1 -> Function[{x, k, h}, Sinc[\[Pi] (x - k h)/h]]

   nabla = Table[ex6, {k, -Nn, Nn}, {j, -Nn, Nn}] /. 
   Indeterminate :> Limit[ex6, k -> j] 

   vx[x_] = x^2/2 /. x -> k h

   vx1 = Table[vx[x], {k, -Nn, Nn}]
   
   vxmat = DiagonalMatrix[vx1]

   A = nabla + vxmat // N
   
   lamda = Eigenvalues[A]

added 654 characters in body

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edited Jun 16, 2019 at 9:17

Mohamed Raafat

23
4

my question is about solving an eigenvalue problem of the Helmholtz equation using sinc approximation [Del]^2E + V \[Del]^2E + V (x) = \[Lambda] E(x) = [Lambda] E and V(x)= X^2 V(x)= X^2 / 2/ 2

I have a problem in calculating the eigenvalues of this system because the eigenvalueseigen values if this system is known to be [Lambda]= \[Lambda]= {1/2 , 3/2 , 5/2}{1/2 , 3/2 , 5/2}

my code is :

Nn = 8
   h = \[Pi]/Sqrt[2 Nn]
   h1 = h

   exactdelta2 = h1^2 D[S1[\[Phi]1[x], j, h1], { \[Phi]1[x], 2}]

   ex5 = exactdelta2 /. x -> \[Psi]1[x] /. \[Phi]1[\[Psi]1[x_]] :> x /. 
  x -> k h1

   ex6 = ex5 /. S1 -> Function[{x, k, h}, Sinc[\[Pi] (x - k h)/h]]

   nabla = Table[ex6, {k, -Nn, Nn}, {j, -Nn, Nn}] /. 
   Indeterminate :> Limit[ex6, k -> j] 

   vx[x_] = x^2/2 /. x -> k h

   vx1 = Table[vx[x], {k, -Nn, Nn}]
   
   vxmat = DiagonalMatrix[vx1]

   A = nabla + vxmat // N
   
   lamda = Eigenvalues[A]

my question is about solving an eigenvalue problem of the Helmholtz equation using sinc approximation \[Del]^2E + V (x) = \[Lambda] E and V(x)= X^2 / 2

I have a problem in calculating the eigenvalues of this system because the eigenvalues if this system is known to be \[Lambda]= {1/2 , 3/2 , 5/2}

my question is about solving an eigenvalue problem of the Helmholtz equation using sinc approximation [Del]^2E + V (x) = [Lambda] E and V(x)= X^2 / 2

I have a problem in calculating the eigenvalues of this system because the eigen values if this system is known to be [Lambda]= {1/2 , 3/2 , 5/2}

my code is :

Nn = 8
   h = \[Pi]/Sqrt[2 Nn]
   h1 = h

   exactdelta2 = h1^2 D[S1[\[Phi]1[x], j, h1], { \[Phi]1[x], 2}]

   ex5 = exactdelta2 /. x -> \[Psi]1[x] /. \[Phi]1[\[Psi]1[x_]] :> x /. 
  x -> k h1

   ex6 = ex5 /. S1 -> Function[{x, k, h}, Sinc[\[Pi] (x - k h)/h]]

   nabla = Table[ex6, {k, -Nn, Nn}, {j, -Nn, Nn}] /. 
   Indeterminate :> Limit[ex6, k -> j] 

   vx[x_] = x^2/2 /. x -> k h

   vx1 = Table[vx[x], {k, -Nn, Nn}]
   
   vxmat = DiagonalMatrix[vx1]

   A = nabla + vxmat // N
   
   lamda = Eigenvalues[A]

added 5 characters in body

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edited Jun 16, 2019 at 9:14

Szabolcs

236.5k
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asked Jun 16, 2019 at 9:11

Mohamed Raafat

23
4

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