In[]:=
a=4;nMax=2;
In[]:=
V
0
=1;
In[]:=
v[x_]:=-2×
2
a
2
π
×
V
0
×1+Cos
2π×x
a
;​​basis[n_,x_]:=Sqrt[1/a]×
2πnx
a

×
Kx

​​​​basisconj[n_,x_]:=Refine[basis[n,x],{Element[K,Reals],Element[x,Reals],Element[n,Integers]}]
In[]:=
hMatrix=Table[
2
(2n+γ)
KroneckerDelta[m,n]+Integrate[basisconj[m,x]v[x]basis[n,x],{x,0,a}],{m,0,nMax},{n,0,nMax}];
In[]:=
hMatrix//MatrixForm
Out[]//MatrixForm=
-
32
2
π
+
2
γ
-
16
2
π
0
-
16
2
π
-
32
2
π
+
2
(2+γ)
-
16
2
π
0
-
16
2
π
-
32
2
π
+
2
(4+γ)
In[]:=
{eValues,eVectors}=Eigensystem[hMatrix];
In[]:=
N[eValues]//MatrixForm
Out[]//MatrixForm=
0.101321Root[16384-16384
2
π
+2048
4
π
-10240
2
π
γ+3072
4
π
γ-2560
2
π
2
γ
+2304
4
π
2
γ
-64
6
π
2
γ
+768
4
π
3
γ
-96
6
π
3
γ
+96
4
π
4
γ
-52
6
π
4
γ
-12
6
π
5
γ
-
6
π
6
γ
+2560-1280
2
π
+64
4
π
-768
2
π
γ+96
4
π
γ-192
2
π
2
γ
+72
4
π
2
γ
+24
4
π
3
γ
+3
4
π
4
γ
#1+96-20
2
π
-12
2
π
γ-3
2
π
2
γ

2
#1
+
3
#1
&,1]
0.101321Root[16384-16384
2
π
+2048
4
π
-10240
2
π
γ+3072
4
π
γ-2560
2
π
2
γ
+2304
4
π
2
γ
-64
6
π
2
γ
+768
4
π
3
γ
-96
6
π
3
γ
+96
4
π
4
γ
-52
6
π
4
γ
-12
6
π
5
γ
-
6
π
6
γ
+2560-1280
2
π
+64
4
π
-768
2
π
γ+96
4
π
γ-192
2
π
2
γ
+72
4
π
2
γ
+24
4
π
3
γ
+3
4
π
4
γ
#1+96-20
2
π
-12
2
π
γ-3
2
π
2
γ

2
#1
+
3
#1
&,2]
0.101321Root[16384-16384
2
π
+2048
4
π
-10240
2
π
γ+3072
4
π
γ-2560
2
π
2
γ
+2304
4
π
2
γ
-64
6
π
2
γ
+768
4
π
3
γ
-96
6
π
3
γ
+96
4
π
4
γ
-52
6
π
4
γ
-12
6
π
5
γ
-
6
π
6
γ
+2560-1280
2
π
+64
4
π
-768
2
π
γ+96
4
π
γ-192
2
π
2
γ
+72
4
π
2
γ
+24
4
π
3
γ
+3
4
π
4
γ
#1+96-20
2
π
-12
2
π
γ-3
2
π
2
γ

2
#1
+
3
#1
&,3]