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The resolvent norm blows up at an eigenvalue

Example

Assume the setting of Non-invertible elliptic shifts form a discrete set in the self-adjoint case and let $(λ_{k}, e_{k})$ be a weak eigenpair with $∥ e_{k} ∥_{L^{2}} = 1$ (Symmetric elliptic weak eigenpairs). Then for every real $λ \notin {λ_{j}}$ $∥ (L - λ)^{- 1} ∥ \geq \frac{1}{∣ λ_{k} - λ ∣},$ because $(L - λ)^{- 1} e_{k} = e_{k} / (λ_{k} - λ)$ has $L^{2}$ norm $1 / ∣ λ_{k} - λ ∣$ ; combined with the exact formula of the spectral-series corollary this gives $∥ (L - λ)^{- 1} ∥ = 1 / dist (λ, {λ_{j}})$ . Hence the resolvent norm is unbounded on every neighbourhood of an eigenvalue: at $λ = λ_{k} - ε$ , for every sufficiently small $ε > 0$ , it is at least $1 / ε$ . The Fredholm alternative is consistent with this: at $λ = λ_{k}$ the homogeneous problem has the nonzero solution $e_{k}$ , and uniqueness and bounded invertibility both fail.

Facts & Assumptions

Given: the Axiom of Choice and Countable Choice; a bounded open set $Ω \subseteq R^{n}$ ; the symmetric divergence-form operator $L$ with eigenvalues ${λ_{j}}$ and orthonormal eigenbasis; a weak eigenpair $(λ_{k}, e_{k})$ with $∥ e_{k} ∥_{L^{2}} = 1$ ; and a real $λ \notin {λ_{j}}$ .

[F1]

Eigenpair data: $e_{k} \in H_{0}^{1} (Ω)$ and $a (e_{k}, v) = λ_{k} (e_{k}, v)_{L^{2}}$ for all $v$ , equivalently $e_{k} \in D (L)$ and $L e_{k} = λ_{k} e_{k}$ ; the eigenbasis is orthonormal in $L^{2}$ (Symmetric elliptic weak eigenpairs, Discrete spectrum of a symmetric elliptic Dirichlet operator).

[F2]

Real resolvent data: for real $λ \notin {λ_{j}}$ , the base-field operator $L - λ$ is bijective with bounded inverse $R_{λ} = (L - λ)^{- 1} : L^{2} (Ω; K) \to D (L)$ , and $∥ R_{λ} ∥ = 1 / dist (λ, {λ_{j}})$ . In the real case this inverse complexifies to $(\tilde{L} - λ)^{- 1} = - (λ - \tilde{L})^{- 1}$ and has the same norm; thus its complexification is the negative of the library resolvent of $\tilde{L}$ , with the same operator norm (Non-invertible elliptic shifts form a discrete set in the self-adjoint case, The complex $L^{2}$ pairing on equivalence classes, Complex Lp classes and Euclidean test-function conventions, Complexification of a real-linear map, Complexification as $C \otimes_{R} V$ with its canonical real-linear embedding, Resolvent and spectrum of an unbounded operator, The operator norm as the least bound and as the unit-sphere or unit-ball supremum, The space $L^{p} (μ)$ as the quotient by null functions).

Verification

technique · direct

1.1F1F2givenalgebra

Action on the eigenfunction. Since $L e_{k} = λ_{k} e_{k}$ by [F1], for real $λ \neq λ_{k}$ one has $(L - λ) e_{k} = (λ_{k} - λ) e_{k}$ , and applying the inverse $R_{λ}$ of [F2] (which exists because $λ \neq λ_{k}$ and $λ \notin {λ_{j}}$ ) gives $R_{λ} e_{k} = \frac{1}{λ_{k} - λ} e_{k} .$ Taking $L^{2}$ norms and using $∥ e_{k} ∥_{L^{2}} = 1$ , $∥ R_{λ} e_{k} ∥_{L^{2}} = 1 / ∣ λ_{k} - λ ∣$ .

2.1F2step 1.1givenalgebra

Lower bound for the operator norm. By definition of the operator norm, $∥ R_{λ} ∥ \geq ∥ R_{λ} e_{k} ∥_{L^{2}} / ∥ e_{k} ∥_{L^{2}} = 1 / ∣ λ_{k} - λ ∣$ ; combined with the exact formula $∥ R_{λ} ∥ = 1 / dist (λ, {λ_{j}})$ of [F2] the lower bound for this fixed $k$ is an equality exactly when $∣ λ_{k} - λ ∣ = dist (λ, {λ_{j}})$ , that is, when $λ_{k}$ is a nearest eigenvalue.

3.1F1F2step 2.1given∎

Blow-up near an eigenvalue. Fix $k$ and $ε > 0$ such that $λ_{k} - ε \notin {λ_{j}}$ (possible for all sufficiently small $ε$ because the eigenvalue set is discrete); then step 2.1 with $λ = λ_{k} - ε$ gives $∥ R_{λ_{k} - ε} ∥ \geq 1 / ε$ , so the resolvent norm is unbounded on every neighbourhood of $λ_{k}$ . At $λ = λ_{k}$ itself no bounded inverse exists: $e_{k}$ is a nonzero homogeneous solution, so $L - λ_{k}$ is not injective, in agreement with the criterion that $L - λ$ is bijective with bounded inverse exactly for $λ \notin {λ_{j}}$ ; uniqueness and bounded invertibility both fail at an eigenvalue.

Depends on

Used by

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Sources

John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page notes) (standard reference, not scraped)
Leon Simon, Lectures on Partial Differential Equations (Stanford, complete 223-page author scan) (standard reference, not scraped)