Capacity and spectral theory.

*(English)*Zbl 1062.31500
Demuth, Michael (ed.) et al., Spectral theory, microlocal analysis, singular manifolds. Berlin: Akademie Verlag (ISBN 3-05-501776-5/hbk). Math. Top. 14, 12-77 (1997).

From the introduction: It is the purpose of this article to reveal connections between and to give a survey on the results comprising capacity and spectral theory. The article is organized as follows.

In Section 2 we introduce the capacity of a regular Dirichlet form \({\mathcal E}\) and show that for a given Radon measure \(\mu\) the form sum \({\mathcal E}+\mu\) is closable if and only if \(\mu\) charges no sets of zero capacity. At the end of Section 2 we describe several generalizations of the notion of capacity extending the usual definition in the context of a regular Dirichlet form.

The main result of Section 3 is that the Sobolev spaces \(H^1_0(U)\) and \(H^1_0(V)\) coincide if and only if the capacity of the symmetric difference between \(U\) and \(V\) equals zero.

Section 4 is devoted to the study of the behaviour of the discrete spectrum under domain perturbations. We show in particular that the first eigenvalue is shifted if and only if the domain perturbation takes place on a set of positive capacity. In certain special cases it is possible to give an asymptotic expansion of the first eigenvalue in terms of the capacity.

In the last section differences of semigroups of self-adjoint operators \(H_0\), \(H_1\) are considered and there are given trace class conditions on certain “sandwiched” semigroup differences which guarantee the existence and completeness of the corresponding wave operators. It is shown that these conditions are satisfied if \(H_1\) arises from \(H_0\) by imposing Dirichlet boundary conditions on a set of finite capacity. In the special case of the Dirichlet Laplacians on \(L^2(U)\) and \(L^2(V)\) we compare the scattering phases of the corresponding scattering systems and obtain an estimate for the difference of the scattering phases in terms of the capacity of the symmetric difference between \(U\) and \(V\).

Finally in the two appendices on Hunt processes and Dirichlet forms the necessary background knowledge is summarized to make the article accessible for a wide range of readers.

For the entire collection see [Zbl 0882.00015].

In Section 2 we introduce the capacity of a regular Dirichlet form \({\mathcal E}\) and show that for a given Radon measure \(\mu\) the form sum \({\mathcal E}+\mu\) is closable if and only if \(\mu\) charges no sets of zero capacity. At the end of Section 2 we describe several generalizations of the notion of capacity extending the usual definition in the context of a regular Dirichlet form.

The main result of Section 3 is that the Sobolev spaces \(H^1_0(U)\) and \(H^1_0(V)\) coincide if and only if the capacity of the symmetric difference between \(U\) and \(V\) equals zero.

Section 4 is devoted to the study of the behaviour of the discrete spectrum under domain perturbations. We show in particular that the first eigenvalue is shifted if and only if the domain perturbation takes place on a set of positive capacity. In certain special cases it is possible to give an asymptotic expansion of the first eigenvalue in terms of the capacity.

In the last section differences of semigroups of self-adjoint operators \(H_0\), \(H_1\) are considered and there are given trace class conditions on certain “sandwiched” semigroup differences which guarantee the existence and completeness of the corresponding wave operators. It is shown that these conditions are satisfied if \(H_1\) arises from \(H_0\) by imposing Dirichlet boundary conditions on a set of finite capacity. In the special case of the Dirichlet Laplacians on \(L^2(U)\) and \(L^2(V)\) we compare the scattering phases of the corresponding scattering systems and obtain an estimate for the difference of the scattering phases in terms of the capacity of the symmetric difference between \(U\) and \(V\).

Finally in the two appendices on Hunt processes and Dirichlet forms the necessary background knowledge is summarized to make the article accessible for a wide range of readers.

For the entire collection see [Zbl 0882.00015].

##### MSC:

31C15 | Potentials and capacities on other spaces |

31C25 | Dirichlet forms |

47A40 | Scattering theory of linear operators |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |

47D07 | Markov semigroups and applications to diffusion processes |

60J45 | Probabilistic potential theory |