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Eigenfunctions for distinct symmetric elliptic eigenvalues are $L^{2}$ -orthogonal

Statement

Assume Countable Choice. In the symmetric case of The $L^{2}$ operator associated with a symmetric elliptic form, let $(λ, u)$ and $(μ, v)$ be weak eigenpairs as in Symmetric elliptic weak eigenpairs with $λ \neq μ$ . Then $(u, v)_{L^{2}} = 0$ . If the scalar field is $C$ and the coefficients are real, conjugation preserves weak eigenpairs at the same eigenvalue; each nonzero real or imaginary part of an eigenfunction is then a real-valued weak eigenfunction, so an eigenfunction can be chosen real.

Facts & Assumptions

Given: Countable Choice; the symmetric divergence-form case of The $L^{2}$ operator associated with a symmetric elliptic form; weak eigenpairs $(λ, u)$ and $(μ, v)$ with $λ \neq μ$ and $u, v \neq 0$ .

[F1]

Weak eigenpair equations: $a (u, w) = λ (u, w)_{L^{2}}$ and $a (v, w) = μ (v, w)_{L^{2}}$ for every $w \in H_{0}^{1} (Ω)$ , with real $λ, μ$ (Symmetric elliptic weak eigenpairs).

[F2]

Symmetry: $a (u, v) = \overline{a (v, u)}$ for all arguments, and $a (w, w)$ is real (Bounded, coercive and symmetric sesquilinear forms, The $L^{2}$ operator associated with a symmetric elliptic form).

[F3]

Conjugation: for real coefficients the form satisfies $a (\overline{u}, \overline{v}) = \overline{a (u, v)}$ . With the inner product linear in its first argument, $(\overline{u}, w)_{L^{2}} = \overline{(u, \overline{w})_{L^{2}}}$ ; conjugation also preserves $H_{0}^{1} (Ω)$ (Real and imaginary parts, complex conjugation, and modulus, The $L^{2}$ operator associated with a symmetric elliptic form, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1F1F2givenalgebra

Test the eigenequation of $(λ, u)$ at $w = v$ and that of $(μ, v)$ at $w = u$ : [F1] gives $λ (u, v)_{L^{2}} = a (u, v)$ and $μ (v, u)_{L^{2}} = a (v, u)$ . Conjugating the second identity and using symmetry [F2], $μ \overline{(v, u)_{L^{2}}} = \overline{a (v, u)} = a (u, v)$ ; since $\overline{(v, u)_{L^{2}}} = (u, v)_{L^{2}}$ and $λ, μ$ are real, comparison gives $λ (u, v)_{L^{2}} = μ (u, v)_{L^{2}}$ , that is $(λ - μ) (u, v)_{L^{2}} = 0$ . As $λ \neq μ$ and the scalar field is $R$ or $C$ , $(u, v)_{L^{2}} = 0$ .

2.1F1F3step 1.1givenalgebra∎

Real coefficients. Suppose the scalar field is $C$ and the coefficients $a^{i j}, c$ are real (with $b = 0$ ). For $w \in H_{0}^{1} (Ω)$ , [F3] gives $a (\overline{u}, w) = \overline{a (u, \overline{w})} = \overline{λ (u, \overline{w})_{L^{2}}} = λ (\overline{u}, w)_{L^{2}}$ , so $\overline{u}$ is a weak eigenfunction with eigenvalue $λ$ . By linearity, each nonzero one of $Re u = \frac{1}{2} (u + \overline{u})$ and $Im u = \frac{1}{2 i} (u - \overline{u})$ is a real-valued weak eigenfunction at $λ$ ; since $u \neq 0$ , at least one is nonzero, so an eigenfunction can be chosen real. The orthogonality conclusion of step 1.1 is independent of this representative remark.

Depends on

Used by

Elliptic Fredholm solvability can fail at an eigenvalue Counterexample
Higher eigenvalues by orthogonality-constrained minimisation Theorem

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Sources

Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page notes) (standard reference, not scraped)
Richard S. Laugesen, Spectral Theory of Partial Differential Equations (University of Illinois lecture notes, arXiv:1203.2344, complete 120 pages) (standard reference, not scraped)

Eigenfunctions for distinct symmetric elliptic eigenvalues are L2L2-orthogonal