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Spektraltheorie Selbstadjungierter Operatoren Im Hilbertraum Und Elliptischer Differentialoperatoren

Spektraltheorie Selbstadjungierter Operatoren Im Hilbertraum Und Elliptischer Differentialoperatoren

          
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About the Book

In diesem Lehrbuch wird der Spektralsatz für selbstadjungierte Operatoren aus dem Resultat der Linearen Algebra über die Diagonalisierung Hermitescher Matrizen hergeleitet. Dabei werden Lebesgue-Stieltjes-Integrale verwendet und der Auswahl- sowie der Konvergenzsatz von Helly über monotone Funktionen bereitgestellt.

Wir konstruieren die Spektralschar durch eine technisch aufwändige Approximation, wobei die Stieltjes-Umkehrformel im Zentrum des Beweises steht. Ein Ergebnis hiervon ist, dass selbstadjungierte Operatoren nicht nur ein diskretes, sondern auch ein kontinuierliches Spektrum besitzen. Die auftretenden Streueigenwerte können hierbei nicht durch Variationsmethoden gewonnen werden.

Dann wenden wir uns der zentralen Frage zu, welche elliptischen Differentialoperatoren eine selbstadjungierte Fortsetzung besitzen und somit im Geltungsbereich des Spektralsatzes liegen. Hier unterscheiden wir zwischen stabilen elliptischen Differentialoperatoren auf beschränkten Gebieten und denen auf dem ganzen Raum, wie etwa dem Schrödingeroperator. Auch Laplace-Beltrami-Operatoren und der Schwarzsche Operator für Minimalflächen werden im obigen Sinne als selbstadjungiert erkannt.

Am Ende dieses Buches geben wir eine Einführung in die Störungstheorie selbstadjungierter Operatoren. Hier weisen wir die analytische Abhängigkeit der Spektralschar vom Störungsparameter nach.

Dieses Werk zur Spektraltheorie ist insbesondere für das fortgeschrittene Mathematik- und Physikstudium geeignet, Kenntnisse in der Funktionalanalysis und der Theorie elliptischer Differentialgleichungen werden vorausgesetzt.


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Product Details
  • ISBN-13: 9783662580684
  • Publisher: Springer Berlin Heidelberg
  • Binding: Paperback
  • Language: German
  • Returnable: Y
  • Weight: 390 gr
  • ISBN-10: 3662580683
  • Publisher Date: 31 Oct 2018
  • Height: 234 mm
  • No of Pages: 260
  • Spine Width: 15 mm
  • Width: 156 mm


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Spektraltheorie Selbstadjungierter Operatoren Im Hilbertraum Und Elliptischer Differentialoperatoren
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