Applications

7.1 Differential Equations

Example (Spectral Theory and ODEs). Consider the Sturm-Liouville problem $-u"' + q(x)u = \lambda u$ on $[a, b]$ with boundary conditions $u(a) = u(b) = 0$. The inverse operator $T = (-d^2/dx^2 + q)^{-1}$ is a compact self-adjoint operator on $L^2[a, b]$. By the spectral theorem, the eigenfunctions form an orthonormal basis, and the eigenvalues $\lambda_n \to \infty$.

7.2 Quantum Mechanics

In quantum mechanics, the state space of a system is a Hilbert space $H$ (typically $L^2(\mathbb{R}^3)$). Observables are self-adjoint operators on $H$. The spectral theorem guarantees that every observable has a spectral decomposition:

$A = \int_{\sigma(A)} \lambda\, dP(\lambda)$

where $P$ is the projection-valued measure associated with $A$.

Example. The position operator $(X\psi)(x) = x\psi(x)$ and momentum operator $(P\psi)(x) = -i\hbar\psi'(x)$ are self-adjoint on suitable domains of $L^2(\mathbb{R})$. The canonical commutation relation $[X, P] = i\hbar I$ is fundamental to quantum mechanics.

7.3 Weak Solutions of Partial Differential Equations

Classical solutions of PDEs require pointwise differentiability, which is too restrictive for many problems. Functional analysis provides the framework for weak (distributional) solutions.

Consider the Poisson equation $-\Delta u = f$ on a bounded domain $\Omega \subset \mathbb{R}^n$ with $u|_{\partial\Omega} = 0$. Multiply both sides by a smooth test function $\varphi$ with $\varphi|_{\partial\Omega} = 0$ and integrate by parts:

$\int_\Omega \nabla u \cdot \nabla \varphi\, dx = \int_\Omega f \varphi\, dx$

This is the weak formulation: find $u$ in a suitable function space such that the above holds for all test functions $\varphi$.

The natural space for this problem is the Sobolev space $H^1_0(\Omega)$, defined as the completion of $C_c^\infty(\Omega)$ under the norm $\|u\|_{H^1} = \left(\int_\Omega |u|^2 + |\nabla u|^2\, dx\right)^{1/2}$. The Riesz representation theorem applied to the inner product $\langle u, v\rangle_{H^1_0} = \int_\Omega \nabla u \cdot \nabla v\, dx$ yields existence and uniqueness of the weak solution. This is known as the Lax-Milgram theorem:

Theorem 7.1 (Lax-Milgram). Let $H$ be a Hilbert space and $B : H \times H \to \mathbb{R}$ a bounded bilinear form that is coercive, i.e., there exists $c > 0$ such that $B(u, u) \geq c\|u\|^2$ for all $u \in H$. Then for every bounded linear functional $\ell \in H^*$, there exists a unique $u \in H$ such that $B(u, v) = \ell(v)$ for all $v \in H$.

Sobolev spaces $W^{k,p}(\Omega)$ (for $k \geq 0$, $1 \leq p \leq \infty$) generalise this idea: $W^{k,p}$ consists of functions whose weak derivatives up to order $k$ belong to $L^p$. They are Banach spaces, and $W^{k,2} = H^k$ are Hilbert spaces.