Here is a non-comprehensive list of the notation used in the couse, with an emphasis on course-specific concepts, possible sources of confusion, and notation differing from typical physics / engineering notation.
| Greek | alphabet |
|---|---|
| $\Delta$ | Laplace operator ($\text{div grad}$) |
| $\resolvent$ | Resovlent set |
| $\spectralradius$ | Spectral radius |
| $\sigma$ | Spectrum |
| $\Sigma$ | Bottom of the essential spectrum |
$$\def\resolvent{{\rho}} \def\spectralradius{{\varrho}} \def\laplacian{{\Delta}} \def\contour{C} \def\eigenspace{{\mathcal E}} \def\op{\mathcal} \def\opA{{\mathcal A}} \def\opH{{\mathcal H}} \def\hilbert{{\mathscr H}} \def\graph{G} \def\boundedoperators{\mathscr B} \def\bloch{\mathcal B} \def\indicator{{\mathbf 1}} \def\im{\operatorname{Im}} \def\ker{\operatorname{Ker}} \definecolor{noteblue}{RGB}{123, 145, 178} \definecolor{warnyellow}{RGB}{165, 159, 116} \definecolor{prooftext}{RGB}{85, 85, 85}$$
Function (and other) spaces
| Space | Definition | Hilbert space ? |
|---|---|---|
| $V_0$ | Space $V$ restricted to functions with compact support. | |
| $C^0(\Omega,Y)$ | $\{ f : \Omega \to Y \mid f \text{ is continuous} \}$ | |
| $C^1(\Omega,Y)$ | $\{ f : \Omega \to Y \mid f' \text{ is continuous} \}$ | |
| $C^k(\Omega,Y)$ | $\{ f : \Omega \to Y \mid f \text{ is infinitely differentiable} \}$ | |
| $C^\infty(\Omega,Y)$ | $\{ f : \Omega \to Y \mid f^{(k)} \text{ is continuous} \}$ | |
| $L^2(\Omega)$ | $\{f : \Omega \to \mathbb C \mid \int_{\Omega} \vert f (x) \vert ^2 dx < \infty \}$ | â |
| $L^p(\Omega)$ | $\{f : \Omega \to \mathbb C \mid \int_{\Omega} \vert f (x) \vert ^p dx < \infty \}$ | |
| $L^p_{loc}(\Omega)$ | $\left \{ f : \Omega \rightarrow \mathbb{C} \ \middle \vert \ f\rvert_K \in L^{p}(K) \quad \forall K \in \Omega, K \text { compact} \right \}$ | |
| $L^2_{per} (\Omega)$ | $\{ f \in L^2_{loc} (\mathbb R^3) \vert f \text{ is } \mathbb L \text{ periodic and } \mathbb L \text{ has unit cell } \Omega \}$ | â |
| $L^2_{qp} (\Omega^*, L^2_{per}(\Omega))$ | $\{ \mathbb R^d \ni k \mapsto u_k \in L^2_{per}(\Omega) \vert \int_{\Omega^*} | u_k |^2_{L^2_{per}(Ί)} \ dk < â \ \text{ and } u_{k+G} = u_k e^{-i G â x} \}$ | â |
| $\mathscr L(V)$ | $\{ f : V \to V \mid f \text{ linear} \}$ | |
| $\ell^2(\mathbb C)$ | $\{z : \mathbb N \to \mathbb C \mid \ \sum_{n = 0}^\infty \vert z_i \vert^2 < \infty \}$ | â |
| $\ell^p(\mathbb C)$ | $\{z : \mathbb N \to \mathbb C \mid \ \sum_{n = 0}^\infty \vert z_i \vert^p < \infty \}$ | |
| $H^n(\Omega)$ | $\{ f \in L^2(\Omega) \mid D^\alpha f \in L^2(\Omega) \ \forall \alpha \text{ s.t. } \Vert \alpha \Vert _1 \leq n \}$ | â |
| $H^S_{per} (\Omega)$ | $\{ f \in L^2_{per} (\Omega) \vert \sum_{G \in \mathbb L^*} (1 + \vert G \vert^2)^S \vert \hat f_G \vert^2 < \infty \}$ | â |
Note that $\Omega$ here is used in most cases to denote the set on which the function is defined. However, in the case of periodic function spaces ($L^2_{per}(\Omega), H^S_{per}(\Omega)$), it denotes the unit cell.
| Other | - |
|---|---|
| $\bullet ^*$ | Adjoint (operators) |
| $\tilde \bullet$ | Approximation of $\bullet$ |
| $\overline{\bullet}$ | Complex conjugate (scalars), closure (sets) |
| $\dot \cup$ | Disjoint union |
| $\varnothing$ | Empty set |
| $\indicator_\Omega$ | Indicator function over set $\Omega$ |
| $\langle \bullet,\bullet \rangle$ | Inner product |
| $(\bullet,\bullet)$ | Open interval |
| $[\bullet, \bullet ]$ | Closed interval |
| $\leq$ | Vector subspace (sets), less or equal to (scalars) |
| $\to$ | Strong convergence |
| $\rightharpoonup$ | Weak convergence |
Inner product of Hilbert spaces (for their definitions, see the table above). To obtain the associated norm, recall $\| f \| = \sqrt{\langle f,f \rangle}$.
| Space | Inner Product $\langle f,g \rangle$ |
|---|---|
| $L^2(\Omega)$ | $\int_\Omega \overline{f(x)} g(x) \ dx$ |
| $\ell^2(\mathbb R)$ | $\sum_{i=0}^\infty \overline{f_i} g_i$ |
| $H^n(\Omega)$ | $\sum_{\Vert \alpha \Vert _1 \leq n} \langle D^\alpha f, D^\alpha g \rangle_{L^2}$ |
| $L^2_{per} (\Omega)$ | $\int_{\Omega} \overline{f(x)} g(x) dx$ |
| $L^2_{qp} (\Omega^*, H^1_{per}(\Omega))$ | $\frac1{\vert \Omega^*\vert} \int_{\Omega^*} \langle f_k, g_k \rangle_{L^2_{per}(\Omega)} dk$ |
| Latin | alphabet |
|---|---|
| $A$ | Generic matrix |
| $\opA$ | Generic operator |
| $\bloch$ | Bloch-Floquet transform |
| $\mathbb B$ | Plane wave basis |
| $\mathscr B$ | Set of all bounded operators |
| $\contour$ | Contour in the complex plane |
| $D^\alpha$ | Weak derivative |
| $D(\opA)$ | Domain of $\opA$ |
| $\eigenspace _A(\lambda)$ | Eigenspace of $A$ associated with eigenvalue $\lambda$. |
| $G(\opA)$ | Graph of $\opA$ |
| $H$ | Sobolev space (see function spaces below) |
| $\opH$ | SchrĂśdinger operator / hamiltonian ($- \laplacian / 2 + V$) |
| $\opH_k$ | Bloch fiber |
| $\hilbert$ | Hilbert space |
| $I$ | Identity matrix |
| $\mathbb K$ | $k$-grid or $k$-point mesh |
| $\mathbb L$ | Lattice |
| $\mathbb L^*$ | Reciprocal lattice |
| $q_A(u)$ | Quadratic form ($\langle u, Au \rangle$) |
| $Q(\opA)$ | Form domain of $\opA$ |
| $a_A(u,v)$ | Sesquilinear form ($\langle u, Av \rangle$) |
| $R_A(u)$ | Rayleigh quotient ($\langle u , Au \rangle / \langle u, u \rangle$) |
| $R_z(A)$ | Resolvent ($(A-z I)^{-1}$) |
| $\mathcal T_R$ | Translation operator by $R$ |