Physlib

Physlib.QuantumMechanics.HilbertSpaces.SpaceD.DirichletSubmodule

Dirichlet submodule

i. Overview

In this module we define the Dirichlet submodule of `SpaceDHilbertSpaceOn Ω μ` consisting of equivalence classes of Schwartz maps which vanish on `frontier Ω`. The frontier (or boundary) of a set `Ω` contains all points whose neighborhoods always intersect with both `Ω` and `Ωᶜ`.

These serve as a convenient dense domain for operators acting on wavefunctions satisfying homogeneous Dirichlet boundary conditions on `Ω`.

ii. Key results

- `DirichletSubmoduleOn Ω μ`: The subspace of `SchwartzSubmodule d μ` consisting of Schwartz maps which vanish on the frontier of `Ω`.

iii. Table of contents

  • A. Definitions
  • B. Contained in SchwartzSubmoduleOn
  • C. Density

iv. References

A. Definitions

B. Contained in SchwartzSubmoduleOn

C. Density

4 declarations

definition

C\mathbb{C}-submodule of Schwartz maps vanishing on ∂Ω\partial \Omega

Let Ω\Omega be a set in the dd-dimensional Euclidean space Space d\text{Space } d. The Dirichlet Schwartz map space is the C\mathbb{C}-submodule of the Schwartz space S(Space d,C)\mathcal{S}(\text{Space } d, \mathbb{C}) (the space of rapidly decreasing smooth functions) consisting of all functions ff that vanish on the frontier (boundary) ∂Ω\partial \Omega. That is, the submodule is defined by the carrier set: {f∈S(Space d,C)∣∀x∈∂Ω,f(x)=0} \{f \in \mathcal{S}(\text{Space } d, \mathbb{C}) \mid \forall x \in \partial \Omega, f(x) = 0\}

abbrev

Dirichlet submodule of the Hilbert space L2(Ω,μ)L^2(\Omega, \mu)

Let Ω\Omega be a subset of the dd-dimensional space Space d\text{Space } d and μ\mu be a measure. The Dirichlet submodule is the C\mathbb{C}-submodule of the Hilbert space L2(Ω,μ)L^2(\Omega, \mu) consisting of the equivalence classes of functions that are restrictions of Schwartz functions f∈S(Space d,C)f \in \mathcal{S}(\text{Space } d, \mathbb{C}) vanishing on the frontier ∂Ω\partial \Omega. Formally, it is defined as the image of the submodule of Dirichlet Schwartz maps under the continuous linear map that includes Schwartz functions into the global Hilbert space L2(Space d,μ)L^2(\text{Space } d, \mu) and then projects them onto the restricted Hilbert space L2(Ω,μ)L^2(\Omega, \mu).

theorem

Membership condition for the Dirichlet submodule of L2(Ω,μ)L^2(\Omega, \mu)

Let Ω\Omega be a subset of Space d\text{Space } d and μ\mu be a measure. An element ψ\psi in the Hilbert space L2(Ω,μ)L^2(\Omega, \mu) belongs to the Dirichlet submodule if and only if there exists a Schwartz function f∈S(Space d,C)f \in \mathcal{S}(\text{Space } d, \mathbb{C}) that vanishes on the boundary ∂Ω\partial \Omega such that the projection of ff into L2(Ω,μ)L^2(\Omega, \mu) (via inclusion into the global Hilbert space and subsequent restriction) is equal to ψ\psi.

theorem

DirichletSubmoduleOn Ω μ⊆SchwartzSubmoduleOn Ω μ\text{DirichletSubmoduleOn } \Omega \ \mu \subseteq \text{SchwartzSubmoduleOn } \Omega \ \mu

Let Ω\Omega be a subset of the dd-dimensional space Space d\text{Space } d and μ\mu be a measure. The Dirichlet submodule of the Hilbert space L2(Ω,μ)L^2(\Omega, \mu) (which consists of the equivalence classes of Schwartz functions that vanish on the frontier ∂Ω\partial \Omega) is a submodule of the Schwartz submodule of L2(Ω,μ)L^2(\Omega, \mu) (which consists of the equivalence classes of all Schwartz functions restricted to Ω\Omega).