Functional Analysis: Spectral Theory¶

Job: develop the spectral theory of unbounded operators at frontier depth — self-adjoint extensions via deficiency indices (von Neumann), the Friedrichs extension of semibounded operators, the projection-valued spectral measure and Stone's theorem, Weyl's theorem on the essential spectrum, the HVZ theorem for Schrödinger operators, and the KLMN form methods for perturbation theory.

Core¶

Self-adjoint extensions and deficiency indices¶

A densely defined operator $T:\mathrm{Dom}(T)\subset H\to H$ is symmetric if $T\subset T^*$ (i.e. $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all $x,y\in\mathrm{Dom}(T)$, with $T^*$ the adjoint on the maximal domain $\mathrm{Dom}(T^*)=\{y:\langle T\cdot,y\rangle=\langle x,T^\ast y\rangle\text{ for some }T^\ast y\}$). $T$ is self-adjoint if $T=T^*$ (i.e. $T$ symmetric and $\mathrm{Dom}(T)=\mathrm{Dom}(T^*)$), and essentially self-adjoint if its closure $\overline T$ is self-adjoint (equivalently, $T$ has a unique self-adjoint extension).

Theorem (von Neumann extension theory). For a closed symmetric operator $T$, the deficiency subspaces are $N_\pm=\ker(T^*\mp i)$, the deficiency indices are $n_\pm=\dim N_\pm$. The self-adjoint extensions of $T$ are in bijection with the unitary maps $U:N_+\to N_-$: for each such $U$, the domain $$ \mathrm{Dom}(T_U)=\mathrm{Dom}(T^*)\oplus{(I-U)\phi\mid\phi\in N_+} $$ defines a self-adjoint extension $T_U$ of $T$. In particular, self-adjoint extensions exist iff $n_+=n_-$, and the extension is unique ($T$ essentially self-adjoint) iff $n_+=n_-=0$.

Property	Domain condition	Deficiency indices
Symmetric	$T\subset T^*$ (the form identity holds on $\mathrm{Dom}(T)$)	arbitrary $n_\pm$
Self-adjoint	$T=T^$ ($\mathrm{Dom}(T)=\mathrm{Dom}(T^)$)	$n_+=n_-=0$
Essentially self-adjoint	$\overline T$ self-adjoint ($T$ has a unique self-adjoint extension)	$n_+=n_-=0$
Has self-adjoint extensions	$n_+=n_-$ (the extensions parametrized by $U:N_+\to N_-$)	$n_+=n_-$
No self-adjoint extension	$n_+\ne n_-$	$n_+\ne n_-$

The Friedrichs extension¶

Theorem (Friedrichs). Let $T$ be a semibounded symmetric operator, $T\ge c$ (i.e. $\langle Tx,x\rangle\ge c\|x\|^2$ for all $x\in\mathrm{Dom}(T)$). Then $T$ admits a distinguished self-adjoint extension $T_F$, the Friedrichs extension, with $\mathrm{Dom}(T_F)\subset\mathrm{Dom}(t)$ where $t$ is the closed quadratic form associated to $T$, and $T_F\ge c$. The Friedrichs extension is the unique self-adjoint extension whose domain is contained in the form domain; for elliptic operators on bounded domains, $T_F$ corresponds to the Dirichlet boundary condition.

The Friedrichs extension is the canonical self-adjoint extension for semibounded operators (elliptic operators, Schrödinger operators $-\Delta+V$ with $V$ bounded below) — it is the one used in the variational / spectral theory (the eigenvalues $\lambda_k$ are the min-max values of the Rayleigh quotient on the form domain).

Spectral measures and the spectral theorem¶

Theorem (spectral theorem, unbounded self-adjoint). For a self-adjoint operator $T$ on a Hilbert space $H$, there is a unique projection-valued measure $E$ on the Borel $\sigma$-algebra of $\mathbb{R}$ with values in orthogonal projections on $H$ such that $$ T=\int_{\mathbb{R}}\lambda\,dE(\lambda),\qquad \mathrm{Dom}(T)=\Bigl{x\in H:\int\lambda^2\,d|E(\lambda)x|^2<\infty\Bigr}. $$ The functional calculus $f(T)=\int f\,dE$ is defined for Borel $f$ on the domain $\mathrm{Dom}(f(T))=\{x:\int|f|^2\,d\|Ex\|^2<\infty\}$.

Case	Structure
Bounded self-adjoint	$T=\int_{\sigma(T)}\lambda\,dE(\lambda)$ with $\sigma(T)\subset\mathbb{R}$ compact; continuous functional calculus $C(\sigma(T))\to B(H)$
Unbounded self-adjoint	$T=\int_\mathbb{R}\lambda\,dE(\lambda)$ with $\mathrm{Dom}(T)$ as above; Borel functional calculus
Normal (bounded / unbounded)	$T=\int_\mathbb{C}\lambda\,dE(\lambda)$, $TT^=T^T$ on $\mathrm{Dom}(T)=\mathrm{Dom}(T^*)$
Unitary	$U=\int_{S^1}\lambda\,dE(\lambda)$, the spectral measure on the circle

Theorem (Stone). A map $U:\mathbb{R}\to B(H)$, $t\mapsto U(t)$, is a strongly continuous one-parameter unitary group ($U(t+s)=U(t)U(s)$, $U(0)=I$, $t\mapsto U(t)x$ continuous for each $x$) iff there is a unique self-adjoint operator $T$ (the generator) with $U(t)=e^{itT}$ (via the functional calculus). This is the bridge to quantum mechanics (the Schrödinger equation $i\partial_t\psi=T\psi$ with self-adjoint Hamiltonian $T$) and to PDE (the wave / Schrödinger / heat propagator).

Essential spectrum and Weyl's theorem¶

The essential spectrum $\sigma_{\mathrm{ess}}(T)$ is the set of $\lambda\in\sigma(T)$ that are not isolated eigenvalues of finite multiplicity; equivalently, $\lambda\in\sigma_{\mathrm{ess}}(T)$ iff $T-\lambda I$ is not a Fredholm operator (the range not closed, or the kernel / cokernel infinite-dimensional).

Theorem (Weyl). For a self-adjoint $T$ and a compact operator $K$ (or $T$-relatively compact), $\sigma_{\mathrm{ess}}(T+K)=\sigma_{\mathrm{ess}}(T)$. The essential spectrum is invariant under compact perturbations.

Theorem (HVZ, Hunziker–van Winter–Zhislin). For the Schrödinger operator $H=-\Delta+V$ on $L^2(\mathbb{R}^n)$ with $V(x)\to 0$ as $|x|\to\infty$ (and mild regularity), $\sigma_{\mathrm{ess}}(H)=[0,\infty)$. The essential spectrum is the continuous spectrum of the free Laplacian $-\Delta$, and the discrete eigenvalues (bound states) lie below $0$ (or in the gaps, for more general $V$).

Form methods and the KLMN theorem¶

Theorem (Kato representation theorem). For a semibounded self-adjoint operator $T\ge c$ on $H$, there is a unique closed semibounded quadratic form $t$ with $t[x,y]=\langle Tx,y\rangle$ on $\mathrm{Dom}(T)$ and $\mathrm{Dom}(t)$ the form domain; conversely, every closed semibounded form $t$ is represented by a unique self-adjoint operator $T\ge c$ with $\mathrm{Dom}(T)\subset\mathrm{Dom}(t)$. This is the form-operator correspondence (the Friedrichs extension is the operator of the form closure).

Theorem (KLMN, Kato–Lions–Lax–Milgram–Nelson). Let $t_0$ be a closed semibounded form on $H$ (with associated self-adjoint operator $T_0$), and let $t_1$ be a relatively form-bounded perturbation: $|t_1[x,x]|\le a\,t_0[x,x]+b\|x\|^2$ with $a<1$. Then $t=t_0+t_1$ is closed semibounded, and the associated self-adjoint operator $T$ is a self-adjoint extension of $T_0$ on $\mathrm{Dom}(T_0)\cap\mathrm{Dom}(t)$. This is the rigorous framework for defining Schrödinger operators $-\Delta+V$ with singular potentials (the Kato class): the form $t_1[\psi,\psi]=\int V|\psi|^2$ is relatively form-bounded with respect to the Dirichlet form $\int|\nabla\psi|^2$ for $V$ in the Kato class (Kato's inequality), giving a self-adjoint $-\Delta+V$ without requiring $V$ to be a bounded operator perturbation.

See it¶

Self-adjoint extensions of $-\Delta$ on $(0,\infty)$ with domain $C_c^\infty(0,\infty)$. Let $T=-\Delta|_{C_c^\infty(0,\infty)}$ (the symmetric operator on the half-line). The adjoint $T^*=-\Delta|_{\mathrm{Dom}(T^*)}$ with $\mathrm{Dom}(T^*)=\{u\in L^2(0,\infty):u,u''\in L^2,\;u\text{ in the distributional sense}\}$ (no boundary condition at $0$). The deficiency subspaces:

\[ N_\pm=\ker(T^*\mp i)=\{u\in L^2(0,\infty):u''=\pm i u,\;u\in L^2\}. \]

Solving $u''=iu$ gives $u(x)=Ae^{(1-i)x/\sqrt 2}+Be^{-(1+i)x/\sqrt 2}$; the $L^2(0,\infty)$ condition at $+\infty$ selects the decaying exponential $e^{-(1+i)x/\sqrt 2}$ (up to scalar), so $N_+=\mathbb{C}\,e^{-(1+i)x/\sqrt 2}$ (1-dimensional). Similarly $N_-=\mathbb{C}\,e^{(-1+i)x/\sqrt 2}$. Hence the deficiency indices are $n_+=n_-=1$ (the $(1,1)$ case), and by von Neumann's theorem the self-adjoint extensions are parametrized by $U(1)=S^1$ (the unitary maps $N_+\to N_-$, parametrized by a phase $e^{i\theta}$). Concretely, the extensions are the boundary conditions at $0$:

\[ \mathrm{Dom}(T_\alpha)=\{u\in H^2(0,\infty):u'(0)=\alpha\,u(0)\},\qquad\alpha\in\mathbb{R}\cup\{\infty\}, \]

with $\alpha=\infty$ the Dirichlet ($u(0)=0$, the Friedrichs extension, the canonical semibounded extension since $-\Delta\ge 0$) and $\alpha\in\mathbb{R}$ the Robin boundary conditions (a 1-parameter family of self-adjoint extensions). The essential spectrum is $[0,\infty)$ for all extensions (by Weyl: $-\Delta$ on the half-line with any local boundary condition at $0$ differs from the free Laplacian on $\mathbb{R}$ by a compact / local perturbation, and the free essential spectrum is $[0,\infty)$). The point spectrum (discrete eigenvalues, the bound states) depends on $\alpha$: for $\alpha<0$ there is a single negative eigenvalue $\lambda=-\alpha^2$ (the bound state $\psi(x)=e^{-\alpha x}$, normalizable for $\alpha<0$), while for $\alpha\ge 0$ there are no bound states. Hence the 1-parameter family of self-adjoint extensions all share the essential spectrum $[0,\infty)$ (Weyl invariance) but differ in the discrete spectrum (the boundary condition controls the bound states) — the operational content of the von Neumann extension theory and Weyl's theorem on a concrete example.

Check¶

Compute the deficiency indices of $-\Delta$ on $(0,\infty)$ with domain $C_c^\infty(0,\infty)$, and identify the parametrization of the self-adjoint extensions.

The deficiency subspaces are $N_\pm=\ker(T^*\mp i)=\{u\in L^2(0,\infty):u''=\pm iu\}$; solving the ODE and imposing $L^2(0,\infty)$ at $+\infty$ gives one-dimensional $N_\pm$ (spanned by the decaying exponentials $e^{-(1\pm i)x/\sqrt 2}$), so $n_+=n_-=1$. By von Neumann's theorem, the self-adjoint extensions are in bijection with the unitary maps $U:N_+\to N_-$, i.e. $U(1)=S^1$ (parametrized by a phase $e^{i\theta}\in S^1$). Concretely, the extensions correspond to the boundary conditions $u'(0)=\alpha u(0)$ for $\alpha\in\mathbb{R}$ (with $\alpha=\infty$ for the Dirichlet $u(0)=0$, the Friedrichs extension), a 1-parameter family indexed by $\alpha\in\mathbb{R}\cup\{\infty\}\cong S^1$ (via the stereographic / Cayley identification of $\alpha$ with $e^{i\theta}$). The deficiency indices $(1,1)$ reflect the single boundary condition at $0$ that can be imposed (the boundary is 0-dimensional, the deficiency indices match the number of boundary conditions).

Explain why the Friedrichs extension is the Dirichlet extension for $-\Delta$ on a bounded domain, and identify the form domain in this case.

The symmetric operator $T=-\Delta|_{C_c^\infty(\Omega)}$ on a bounded domain $\Omega$ is semibounded ($T\ge 0$ since $\langle -\Delta u,u\rangle=\|\nabla u\|_{L^2}^2\ge 0$). The associated quadratic form is $t[u,u]=\int|\nabla u|^2$ on $C_c^\infty(\Omega)$; its closure has form domain $\mathrm{Dom}(t)=H^1_0(\Omega)$ (the closure of $C_c^\infty$ in the $H^1$ norm, i.e. the Sobolev space with zero trace on $\partial\Omega$). The Friedrichs extension $T_F$ is the self-adjoint operator associated to the closed form $t$, with $\mathrm{Dom}(T_F)\subset H^1_0(\Omega)$; concretely $T_F=-\Delta|_{\mathrm{Dom}(T_F)}$ with $\mathrm{Dom}(T_F)=H^2(\Omega)\cap H^1_0(\Omega)$ (the elliptic regularity gives $H^2$ for smooth $\partial\Omega$), which is exactly the Dirichlet Laplacian ($u|_{\partial\Omega}=0$). The form domain $H^1_0(\Omega)$ encodes the Dirichlet boundary condition (zero trace), so the Friedrichs extension is the Dirichlet extension — the canonical semibounded self-adjoint extension for the Laplacian on a bounded domain. Other extensions (Robin / Neumann) correspond to different form domains (e.g. $H^1(\Omega)$ for the Neumann, no zero-trace condition).

Pitfalls¶

Symmetric $\ne$ self-adjoint for unbounded operators; domains are part of the data: a symmetric $T\subset T^*$ may fail to be self-adjoint ($\mathrm{Dom}(T)\subsetneq\mathrm{Dom}(T^*)$); the deficiency indices $n_\pm$ measure the failure, and self-adjoint extensions exist iff $n_+=n_-$. Treating symmetric as self-adjoint is the standard error (and prevents the spectral theorem).
The Friedrichs extension is the canonical semibounded extension but not the only one: for a semibounded $T$ with $n_+=n_-$, there is a family of self-adjoint extensions (parametrized by $U:N_+\to N_-$), and the Friedrichs is the one whose domain is contained in the form domain. Other extensions may have different spectral / boundary-condition properties.
The essential spectrum is invariant under compact perturbations (Weyl) but not under relatively form-bounded perturbations: the KLMN theorem handles form-bounded perturbations and gives a self-adjoint operator, but the essential spectrum may shift (e.g. $-\Delta+V$ with $V\not\to 0$ at infinity can have a different essential spectrum than $-\Delta$).
The HVZ theorem requires $V\to 0$ at infinity: the essential spectrum $[0,\infty)$ for $-\Delta+V$ needs $V(x)\to 0$; for confining potentials $V\to+\infty$ the essential spectrum is empty (pure point), for periodic $V$ the essential spectrum is a band structure.
The KLMN theorem requires the relative bound $a<1$: for $a\ge 1$ the form may fail to be closed / semibounded, and the KLMN theorem does not apply; the form-sum may need renormalization (e.g. singular potentials beyond the Kato class).
Stone's theorem requires strong continuity (not norm continuity): a strongly continuous unitary group $U(t)$ has a self-adjoint generator $T$; a norm-continuous unitary group has a bounded generator (a much stronger condition, and the generator is then in $B(H)$).

Refs¶

Reed–Simon, Methods of Modern Mathematical Physics II (Fourier analysis, self-adjointness); I (functional analysis)
Kato, Perturbation Theory for Linear Operators (the Kato representation, KLMN, Kato class)
Teschl, Mathematical Methods in Quantum Mechanics; Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space
Hislop–Segal, Introduction to Spectral Theory (HVZ, essential spectrum); Cycon–Froese–Kirsch–Simon, Schrödinger Operators

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Property	Domain condition	Deficiency indices
Symmetric	\(T\subset T^*\) (the form identity holds on \(\mathrm{Dom}(T)\))	arbitrary \(n_\pm\)
Self-adjoint	\(T=T^\) (\(\mathrm{Dom}(T)=\mathrm{Dom}(T^)\))	\(n_+=n_-=0\)
Essentially self-adjoint	\(\overline T\) self-adjoint (\(T\) has a unique self-adjoint extension)	\(n_+=n_-=0\)
Has self-adjoint extensions	\(n_+=n_-\) (the extensions parametrized by \(U:N_+\to N_-\))	\(n_+=n_-\)
No self-adjoint extension	\(n_+\ne n_-\)	\(n_+\ne n_-\)