On the estimations of the small eigenvalues of Sturm–Liouville operators with periodic and antiperiodic boundary conditions

Nur, Cemile

doi:10.1186/s13661-018-1111-y

Boundary Value Problems

Table 1 Estimations of eigenvalues for \(q(x)=2\cos(2\pi x)\)

From: On the estimations of the small eigenvalues of Sturm–Liouville operators with periodic and antiperiodic boundary conditions

\(x_{n,i}\)

\(\vert x_{n,i+1}-x_{n,i} \vert \)

\(y_{n,i}\)

\(\vert y_{n,i+1}-y_{n,i} \vert \)

±1

0
1
2
3
4
5

1.000000000000
39.472044558692
39.469974401006
39.469974548582
39.469974548572
39.469974548572

38.472044558692
0.002070157686
0.000000147576
0.000000000011
0.000000000000

1.000000000000
39.478417489727
39.478417378628
39.478417378628

38.478417489727
0.000000111099
0.000000000000

±2

0
1
2
3
4
5

1.000000000000
157.911370091955
157.908604814554
157.908604885553
157.908604885551
157.908604885551

156.911370091955
0.002765277401
0.000000070999
0.000000000002
0.000000000000

1.000000000000
157.914499110787
157.913670814745
157.913670814754
157.913670814754

156.914499110787
0.000828296043
0.000000000009
0.000000000000

±3

0
1
2
3
4
5

1.000000000000
355.299384450125
355.302139969517
355.302139933433
355.302139933434
355.302139933434

354.299384450125
0.002755519393
0.000000036084
0.000000000001
0.000000000000

1.000000000000
355.304171727348
355.310824599657
355.310824428906
355.310824428910
355.310824428910

354.304171727348
0.006652872309
0.000000170751
0.000000000004
0.000000000000

±4

0
1
2
3
4
5

1.000000000000
631.651859197264
631.651867232362
631.651867232298
631.651867232298

630.651859197264
0.000008035097
0.000000000064
0.000000000000

1.000000000000
631.653667378752
631.658300314310
631.658300253644
631.658300253645
631.658300253645

630.653667378752
0.004632935559
0.000000060666
0.000000000001
0.000000000000

±5

0
1
2
3
4
5

1.000000000000
986.958854447771
986.958137370115
986.958137373917
986.958137373917
986.958137373917

985.958854447771
0.000717077657
0.000000003802
0.000000000000
0.000000000000

1.000000000000
986.959735986823
986.963254598781
986.963254570909
986.963254570909
986.963254570909

985.959735986823
0.003518611957
0.000000027872
0.000000000000
0.000000000000

±6

0
1
2
3
4

1.000000000000
1421.222019515655
1421.221085279699
1421.221085283246
1421.221085283246

1420.222019515655
0.000934235956
0.000000003547
0.000000000000

1.000000000000
1421.222516542946
1421.225336517065
1421.225336502112
1421.225336502112

1420.222516542946
0.002819974119
0.000000014953
0.000000000000

±7

0
1
2
3
4

1.000000000000
1934.441758498203
1934.440773930878
1934.440773933686
1934.440773933686

1933.441758498203
0.000984567325
0.000000002808
0.000000000000

1.000000000000
1934.442066670380
1934.444411101068
1934.444411092167
1934.444411092167

1933.442066670380
0.002344430688
0.000000008901
0.000000000000

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