We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
from Part Two - The Laplace and Schrödinger Operators
Published online by Cambridge University Press: 03 November 2022
We discuss the definition of the Laplace operator on an open subset in Euclidean space as a self-adjoint operator with Dirichlet or Neumann boundary conditions and we derive its basic spectral properties. Among others, we include the spectral inequalities of Faber–Krahn, Hersch, and Friedlander. Then, using the technique of Dirichlet–Neumann bracketing, we derive Weyl's law for the asymptotic distribution of eigenvalues. We supplement this with a discussion of non-asymptotic bounds, including Pólya's conjecture and its proof for tiling domains and domains of product form. We present the sharp eigenvalue bounds of Berezin and Li–Yau. Finally, using separation of variables in spherical coordinates, we discuss the Laplacian on a ball.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
