## Linear Operators: Spectral theory |

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Page 1433

is called the indicial

roots ei, ..., ea, no two of which differ by an integer, then the set of solutions of [*]

has a basis of the form o,(z) = 2*p,(z), where p, is analytic and non-zero in the ...

is called the indicial

**equation**of [+] at zero. If the indicial**equation**has distinctroots ei, ..., ea, no two of which differ by an integer, then the set of solutions of [*]

has a basis of the form o,(z) = 2*p,(z), where p, is analytic and non-zero in the ...

Page 1527

To deal with the confluent hypergeometric

some of the theory of

Lj = (d/dz)”f+p(z)(d/dz)f-i-q(z)f = 0 to have an irregular singularity of order (or type)

...

To deal with the confluent hypergeometric

**equation**we must consequently usesome of the theory of

**equations**with ... It is most convenient to take the**equation**Lj = (d/dz)”f+p(z)(d/dz)f-i-q(z)f = 0 to have an irregular singularity of order (or type)

...

Page 1528

The first of these algebraic

of the differential

differential

irregular ...

The first of these algebraic

**equations**, which is simply the characteristic**equation**of the differential

**equation**, ... + o-1)2)}z-of, f being a solution of the originaldifferential

**equation**Lj = 0, we find that L'f' has rational coefficients, and anirregular ...

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

Copyright | |

38 other sections not shown

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero