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Table 2 Estimations of eigenvalues for \(q(x)=2\cos(2\pi x)+2\cos(4\pi x)\)

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

n

i

\(x_{n,i}\)

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

\(y_{n,i}\)

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

±1

0

1

2

3

4

5

1.000000000000

38.473844750118

38.472688962751

38.472689008589

38.472689008587

38.472689008587

37.473844750118

0.001155787367

0.000000045838

0.000000000002

0.000000000000

1.000000000000

40.423036168470

40.471787835885

40.471786569354

40.471786569387

40.471786569387

39.423036168470

0.048751667415

0.000001266531

0.000000000033

0.000000000000

±2

0

1

2

3

4

5

1.000000000000

157.708686755063

157.906540179470

157.906534021274

157.906534021465

157.906534021465

156.708686755063

0.197853424408

0.000006158197

0.000000000192

0.000000000000

1.000000000000

157.910954517251

157.913719642989

157.913719639257

157.913719639257

156.910954517251

0.002765125738

0.000000003732

0.000000000000

±3

0

1

2

3

4

5

1.000000000000

355.274676914265

355.300564251475

355.300563843658

355.300563843665

355.300563843665

354.274676914265

0.025887337209

0.000000407817

0.000000000006

0.000000000000

1.000000000000

355.304635333209

355.313997921297

355.313997585704

355.313997585716

355.313997585716

354.304635333209

0.009362588088

0.000000335592

0.000000000012

0.000000000000

±4

0

1

2

3

4

5

1.000000000000

631.645567421755

631.650602854392

631.650602806214

631.650602806214

631.650602806214

630.645567421755

0.005035432636

0.000000048178

0.000000000000

0.000000000000

1.000000000000

631.653044860730

631.660413259797

631.660413130168

631.660413130170

631.660413130170

630.653044860730

0.007368399068

0.000000129629

0.000000000002

0.000000000000

±5

0

1

2

3

4

1.000000000000

986.956043584523

986.957082783600

986.957082776914

986.957082776914

985.956043584523

0.001039199076

0.000000006685

0.000000000000

1.000000000000

986.959230332354

986.964838571940

986.964838513367

986.964838513368

986.964838513368

985.959230332354

0.005608239586

0.000000058572

0.000000000001

0.000000000000

±6

0

1

2

3

4

1.000000000000

1421.220438496119

1421.220181024587

1421.220181025778

1421.220181025777

1421.220181025777

1420.220438496119

0.000257471532

0.000000001190

0.000000000001

0.000000000000

1.000000000000

1421.222125234640

1421.226603425567

1421.226603394604

1421.226603394604

1421.226603394604

1420.222125234640

0.004478190927

0.000000030964

0.000000000000

0.000000000000

±7

0

1

2

3

4

1.000000000000

1934.440746482760

1934.439982571624

1934.439982574285

1934.439982574285

1933.440746482760

0.000763911135

0.000000002660

0.000000000000

1.000000000000

1934.441756076886

1934.445466740559

1934.445466722323

1934.445466722323

1933.441756076886

0.003710663673

0.000000018236

0.000000000000