The Spectral Theory of Periodic Differential Equations

Title Page......Page 1
Copyright Page......Page 2
Preface\0......Page 3
Contents......Page 5
1.1 Floquet theory\0......Page 7
1.2 Hill\'s equation\0......Page 11
1.3 Boundedness and periodicity of solutions\0......Page 14
1.4 Complex-valued coefficients\0......Page 17
1.5 Systems of differential equations\0......Page 18
1.6 Systems all of whose solutions are periodic\0......Page 21
Notes and references\0......Page 23
2.1 Introduction\0......Page 25
2.2 The periodic and semi-periodic eigenvalue problems\0......Page 26
2.3 The function D(A)\0......Page 33
2.4 Two further eigenvalue problems\0......Page 37
2.5 The Mathieu equation\0......Page 40
Notes and references\0......Page 42
3.1 Zeros of eigenfunctions\0......Page 43
3.2 Oscillation of solutions\0......Page 47
3.3 Two inequalities concerning A2\0......Page 48
3.4 The right-hand end-points of the stability intervals\0......Page 50
3.5 Stability criteria\0......Page 54
Notes and references\0......Page 56
4.1 Prufer transformation formulae\0......Page 58
4.2 Asymptotic estimates\0......Page 61
4.3 Asymptotic formulae for solutions\0......Page 67
4.4 An improvement of (4.2.10)\0......Page 70
4.5 The length of the instability intervals\0......Page 71
64. Inverse problems\0......Page 75
Notes and references\0......Page 80
5.1 Differential operators\0......Page 82
5.2 Gaps in the essential spectrum\0......Page 85
5.3 The spectrum of L\0......Page 87
5.4 The lengths of the gaps\0......Page 91
5.5 The least eigenvalues\0......Page 95
Notes and references\0......Page 100
6.2 Eigenvalue problems\0......Page 102
6.3 Variational results\0......Page 105
6.4 Differential operators\0......Page 108
6.5 The conditional stability intervals\0......Page 110
6.6 The Bloch functions\0......Page 112
6.7 The reciprocal lattice and Brillouin zone\0......Page 116
6.8 The case q(x) = 0\0......Page 118
6.9 The first stability interval\0......Page 121
6.10 The spectrum\0......Page 123
Notes and references\0......Page 128
References\0......Page 130