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Fourier spectrum and positivity of the periodic lattice Laplacian

Statement

Let $d, L \geq 1$ be integers and let $Λ_{d, L} : = (Z / L Z)^{d}$ , where $Z / L Z$ is the quotient set of integers modulo $L$ . Use the quotient addition $[a]_{L} + [b]_{L} = [a + b]_{L}$ and additive inverse $- [a]_{L} = [- a]_{L}$ , and define these operations coordinatewise on $Λ_{d, L}$ . For $x \in Λ_{d, L}$ , write $\tilde{x} = ({\tilde{x}}_{1}, \dots, {\tilde{x}}_{d}) \in {0, \dots, L - 1}^{d}$ for its unique coordinate representatives. For $1 \leq j \leq d$ , let $e_{j} \in Λ_{d, L}$ have coordinate $[1]_{L}$ in position $j$ and $[0]_{L}$ elsewhere. Count the two terms $x + e_{j}$ and $x - e_{j}$ separately even when they coincide for $L = 1$ or $L = 2$ .

For a real-valued function $f : Λ_{d, L} \to R$ , define

$(- Δ f) (x) : = \sum_{j = 1}^{d} (2 f (x) - f (x + e_{j}) - f (x - e_{j})) .$

For a complex-valued function, use the same formula for $- Δ$ , which is its complex-linear extension. Write $C^{Λ_{d, L}}$ for the complex-valued functions on $Λ_{d, L}$ with pointwise vector operations. For $r = (r_{1}, \dots, r_{d}) \in {0, \dots, L - 1}^{d}$ , define the function $χ_{r} : Λ_{d, L} \to C$ by

$χ_{r} (x) : = \exp ⁣ (\frac{2 π i}{L} \sum_{j = 1}^{d} r_{j} {\tilde{x}}_{j}) .$

Use the inner product $⟨ f, g ⟩ : = L^{- d} \sum_{x \in Λ_{d, L}} f (x) \overline{g (x)}$ . Identify real functions with coordinate vectors using the lexicographic order on the representatives $\tilde{x}$ , and let $I_{Λ_{d, L}}$ be the identity operator on those functions. A real matrix $A$ is symmetric when $A^{T} = A$ ; a real symmetric matrix is positive semidefinite when $v^{T} A v \geq 0$ for every real vector $v$ , and positive definite when $v^{T} A v > 0$ for every nonzero real vector $v$ . For all $r, s \in {0, \dots, L - 1}^{d}$ , the inner products satisfy $⟨ χ_{r}, χ_{s} ⟩ = 1$ if $r = s$ and $0$ otherwise; every complex-valued function on $Λ_{d, L}$ is a unique linear combination of the functions $χ_{r}$ .

For each $r$ , the equation $- Δ χ_{r} = λ_{r} χ_{r}$ holds with $λ_{r} : = 4 \sum_{j = 1}^{d} \sin^{2} (π r_{j} / L)$ ; this scalar is the eigenvalue associated with $χ_{r}$ .

Consequently $- Δ$ is real symmetric positive semidefinite, and for real $m^{2} > 0$ and $κ > 0$ the matrix $m^{2} I_{Λ_{d, L}} + κ (- Δ)$ is real symmetric positive definite with eigenvalues $m^{2} + 4 κ \sum_{j = 1}^{d} \sin^{2} (π r_{j} / L) > 0$ .

Statement references

The congruence class $[a]_{n}$ and the quotient set $Z / n$

For $n \geq 1$ , every class in $Z / n$ has one representative $r$ with $0 \leq r < n$ , so $∣ Z / n ∣ = n$ ; while $Z / 0$ is in bijection with $Z$

Addition and multiplication on $Z / n$ by $[a]_{n} + [b]_{n} = [a + b]_{n}$ and $[a]_{n} [b]_{n} = [a b]_{n}$

For every natural $n$ , $(Z / n, +)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold

Orthogonality of the characters $x \mapsto e^{2 π i k x / N}$ on $Z / N Z$

Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule

The complex exponential by its power series

$\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential

$\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$

$\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$

Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive

The addition formulas for sine and cosine

Pythagorean and parity identities for all six trigonometric functions on their natural domains

Facts & Assumptions

Given: Integers $d, L \geq 1$ , the coordinatewise quotient operations, canonical representatives, and discrete Laplacian of the Statement.

[F1]

Every class in $Z / L Z$ has a unique representative in ${0, \dots, L - 1}$ (The congruence class $[a]_{n}$ and the quotient set $Z / n$ , For $n \geq 1$ , every class in $Z / n$ has one representative $r$ with $0 \leq r < n$ , so $∣ Z / n ∣ = n$ ; while $Z / 0$ is in bijection with $Z$ ).

[F2]

Quotient addition is $[a]_{L} + [b]_{L} = [a + b]_{L}$ and is independent of representatives; $Z / L Z$ with this operation is an additive abelian group, with inverse $- [a]_{L} = [- a]_{L}$ (Addition and multiplication on $Z / n$ by $[a]_{n} + [b]_{n} = [a + b]_{n}$ and $[a]_{n} [b]_{n} = [a b]_{n}$ , For every natural $n$ , $(Z / n, +)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).

[F3]

For $N \geq 1$ and integers $k, ℓ$ , $\sum_{u = 0}^{N - 1} e^{2 π i (k - ℓ) u / N}$ is $N$ if $k \equiv ℓ (m o d N)$ and is $0$ otherwise (Orthogonality of the characters $x \mapsto e^{2 π i k x / N}$ on $Z / N Z$ ).

[F4]

Finite sums over product index sets may be factored and evaluated in either order (Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule).

[F5]

$\exp (z + w) = \exp z \exp w$ and $\exp z = 1$ exactly when $z \in 2 π i Z$ (The complex exponential by its power series, $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ ).

[F6]

For real $t$ , $e^{i t} = \cos t + i \sin t$ and $\cos (2 t) = \cos^{2} t - \sin^{2} t$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The addition formulas for sine and cosine).

[F7]

For real $t$ , $\sin^{2} t + \cos^{2} t = 1$ , $\sin (- t) = - \sin t$ , and $\cos (- t) = \cos t$ (Pythagorean and parity identities for all six trigonometric functions on their natural domains).

[F8]

Complex conjugation fixes real numbers and sends $i$ to $- i$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive); together with [F6] and [F7], this gives $\overline{e^{i t}} = e^{- i t}$ for real $t$ .

Proof

1.1givenF1F2F5algebra

Fix $x \in Λ_{d, L}$ and put $a_{j} = {\tilde{x}}_{j}$ . By [F1]–[F2], the $j$ th representative of $x + e_{j}$ is $a_{j} + 1 - L ϵ_{j}$ , where $ϵ_{j} = 1$ if $a_{j} = L - 1$ and $0$ otherwise; the $j$ th representative of $x - e_{j}$ is $a_{j} - 1 + L η_{j}$ , where $η_{j} = 1$ if $a_{j} = 0$ and $0$ otherwise. All other representatives are unchanged. Substitution into the definition of $χ_{r}$ , followed by the exponential addition law and $\exp (2 π i n) = 1$ for integers $n$ from [F5], gives $χ_{r} (x + e_{j}) = e^{2 π i r_{j} / L} χ_{r} (x)$ and $χ_{r} (x - e_{j}) = e^{- 2 π i r_{j} / L} χ_{r} (x)$ . This also holds for $L = 1$ and $L = 2$ .

1.2F1F3F4F8algebra

For $r, s \in {0, \dots, L - 1}^{d}$ , finite-sum factorization [F4] and the conjugation identity [F8] give $⟨ χ_{r}, χ_{s} ⟩ = \prod_{j = 1}^{d} L^{- 1} \sum_{u = 0}^{L - 1} e^{2 π i (r_{j} - s_{j}) u / L}$ . By [F1], distinct canonical representatives are incongruent modulo $L$ , so [F3] makes each factor $1$ when $r_{j} = s_{j}$ and $0$ otherwise. This is the inner-product identity stated above.

2.1F6F7step 1.1algebra

Substituting the shift identities of step 1.1 into $- Δ$ gives $- Δ χ_{r} = \sum_{j = 1}^{d} (2 - e^{2 π i r_{j} / L} - e^{- 2 π i r_{j} / L}) χ_{r}$ . By [F6]–[F7], each coefficient equals $2 - 2 \cos (2 π r_{j} / L) = 4 \sin^{2} (π r_{j} / L)$ , proving the eigenvalue formula.

2.2F1F3F4F8step 1.2algebra

For any $f : Λ_{d, L} \to C$ , set $\hat{f} (r) = L^{- d} \sum_{y \in Λ_{d, L}} f (y) \overline{χ_{r} (y)}$ . By [F4] and [F8], the frequency sum factors as $\sum_{r} χ_{r} (x) \overline{χ_{r} (y)} = \prod_{j = 1}^{d} \sum_{r_{j} = 0}^{L - 1} e^{2 π i r_{j} ({\tilde{x}}_{j} - {\tilde{y}}_{j}) / L}$ . The canonical-representative theorem [F1] and orthogonality [F3] make this product $L^{d}$ if $x = y$ in $Λ_{d, L}$ and $0$ otherwise. Therefore $\sum_{r} \hat{f} (r) χ_{r} (x) = L^{- d} \sum_{y} f (y) \sum_{r} χ_{r} (x) \overline{χ_{r} (y)} = f (x)$ for every $x$ . If a linear combination $\sum_{r} c_{r} χ_{r}$ vanishes, taking its inner product with $χ_{s}$ and applying step 1.2 gives $c_{s} = 0$ for each $s$ , so the expansion is unique.

3.1givenF2F4step 2.1step 2.2algebra∎

For every real $f$ , the translations $x \mapsto x + e_{j}$ are bijections with inverses $x \mapsto x - e_{j}$ , so reindexing the finite sums [F4] gives $\sum_{x} f (x) (- Δ f) (x) = \sum_{j = 1}^{d} \sum_{x} (f (x + e_{j}) - f (x))^{2} \geq 0$ . Thus $- Δ$ is positive semidefinite; its matrix is symmetric because every shift term is paired with its inverse shift. For nonzero $f$ , $f^{T} (m^{2} I + κ (- Δ)) f = m^{2} \sum_{x} f (x)^{2} + κ \sum_{j = 1}^{d} \sum_{x} (f (x + e_{j}) - f (x))^{2} > 0,$ since $m^{2} > 0$ and $κ > 0$ . By steps 2.1 and 2.2, the spanning functions $χ_{r}$ satisfy the eigenvalue equations in the Statement, so the combined operator has the displayed eigenvalues.

Fourier spectrum and positivity of the periodic lattice Laplacian

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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