18.303: Self-adjointness (reciprocity)and definiteness (positivity) in Green’s functionsS. G. JohnsonOctober 11, 20110 ReviewSuppose we have some vector space V of functions u(x) on a domain Ω, an inner product hu, vi, anda linear operatorˆA. [More specifically, V forms a Sobolev space, in that we require hu,ˆAui to befinite.]ˆA is self-adjoint if hu,ˆAvi = hˆAu, vi for all u, v ∈ V, in which case its eigenvalues λnare realand its eigenfunctions un(x) can be chosen orthonormal.ˆA is positive definite (or semidefinite) ifhu,ˆAui > 0 (or ≥ 0) for all u 6= 0, in which case its eigenvalues are > 0 (or ≥ 0); suppose that weorder them as 0 < λ1≤ λ2≤ ···.Suppose thatˆA is positive definite, so that N(ˆA) = {0} andˆAu = f has a unique solution forall f in some suitable space of functions C(ˆA). Then, for scalar-valued functions u and f , we cantypically writeu(x) =ˆA−1f =ˆx0∈ΩG(x, x0) f (x0) , (1)in terms of a Green’s function G(x, x0), where´x0∈Ωdenotes integration over x0. In this note, wedon’t address how to find G, but instead ask what properties it must have from the self-adjointnessand definiteness ofˆA. [This generalizes in a straightforward way to vector-valued u(x) and f(x), inwhich case G(x, x0) is matrix-valued.]1 Self-adjointness ofˆA−1and reciprocity of GWe can show that (ˆA−1)∗= (ˆA∗)−1, from which it follows that ifˆA =ˆA∗(ˆA is self-adjoint) thenˆA−1is also self-adjoint. In particular, considerˆA−1ˆA = 1: hu, vi = hu,ˆA−1ˆAvi = h(ˆA−1)∗u,ˆAvi =hˆA∗(ˆA−1)∗u, vi, henceˆA∗(ˆA−1)∗= 1 and (ˆA−1)∗= (ˆA∗)−1. And of course, we already knew that theeigenvalues ofˆA−1are λ−1nand the eigenfunctions are un(x).What are the consequences of self-adjointness for G? Suppose the u are scalar functions, andthat the inner product is of the form hu, vi =´Ωw ¯uv for some weight w(x) > 0. From the fact thathu,ˆA−1vi = hˆA−1u, vi, substituting equation (1), we must therefore have:hu,ˆA−1vi =¨x,x0∈Ωw(x)u(x)G(x, x0)v(x0)= hˆA−1u, vi=¨x,x0∈Ωw(x)G(x, x0)u(x0)v(x) =¨x,x0∈Ωw(x0)u(x)G(x0, x)v(x0),where in the last step we have interchanged/relabeled x ↔ x0. Since this must be true for all u and v,it follows thatw(x)G(x, x0) = w(x0)G(x0, x)for all x, x0. This property of G (or its analogues in other systems) is sometimes called reciprocity.In the common case where w = 1 andˆA and G are real (so that the complex conjugation can beomitted), it says that the effect at x from a source at x0is the same as the effect at x0from a source atx.1There are many interesting consequences of reciprocity. For example, its analogue in linearelectrical circuits says that the current at one place created by a voltage at another is the same as ifthe locations of the current and voltage are swapped. Or, for antennas, the analogous theorem saysthat a given antenna works equally well as a transmitter or a receiver.1.1 Example:ˆA = −d2dx2on Ω = [0, L]For this simple example (whereˆA is self-adjoint under hu, vi =´¯uv), with Dirichlet boundaries, wepreviously obtained a Green’s function,G(x, x0) =(1 −x0Lx x < x0(1 −xL)x0x ≥ x0,which obviously obeys the G(x, x0) = G(x0, x) reciprocity relation.2 Positive-definiteness ofˆA−1and positivity of GNot only isˆA−1self-adjoint, but since its eigenvalues are the inverses λ−1nof the eigenvalues ofˆA,then ifˆA is positive-definite (λn> 0) thenˆA−1is also positive-definite (λ−1n> 0). From anotherperspective, ifˆAu = f , then positive-definiteness ofˆA means that 0 < hu,ˆAui = hu, f i = hˆA−1f , f i =h f ,ˆA−1f i for u 6= 0 ⇔ f 6= 0, henceˆA−1is positive-definite. (And ifˆA is a PDE operator withan ascending sequence of unbounded eigenvalues, then the eigenvalues ofˆA−1are a descendingsequence λ−11> λ−12> ··· > 0 that approaches 0 asymptotically from above.1)IfˆA is a real operator (real u give realˆAu), thenˆA−1should also be a real operator (real fgive real u =ˆA−1f ). Furthermore, under fairly general conditions for real positive-definite (elliptic)PDE operatorsˆA, especially for second-derivative (“order 2”) operators, then one can often showG(x, x0) > 0 (except of course for x or x0at the boundaries, where G vanishes for Dirichlet condi-tions).2The analogous fact for matrices A is that if A is real-symmetric positive-definite and it hasoff-diagonal entries ≤ 0 — like our −∇2second-derivative matrices (recall the −1, 2,−1 sequencesin the rows) and related finite-difference matrices — it is called a Stieltjes matrix, and such matricescan be shown to have inverses with nonnegative entries.32.1 Example:ˆA = −∇2with u|∂ Ω= 0Physically, the positive-definite problem −∇2u = f can be thought of as the displacement u inresponse to an applied pressure f , where the Dirichlet boundary conditions correspond to a materialpinned at the edges. The Green’s function G(x, x0) is the limit of the displacement u in responseto a force concentrated at a single point x0. The Green’s function G(x, x0) for some example pointsx0is shown for a 1d domain Ω = [0, 1] in figure 1(left) (a “stretched string”), and for a 2d domainΩ = [−1, 1] × [−1, 1] in figure 1(right) (a “square drum”). As expected, G > 0 everywhere exceptat the edges where it is zero: the whole string/membrane moves in the positive/upwards direction inresponse to a positive/upwards force.1SuchˆA−1integral operators are typically what are called “compact” operators. Functional analysis books often provediagonalizability (a “spectral theorem”) for compact operators first and only later consider diagonalizability of PDE-likeoperators by viewing them as the inverses of compact operators.2See, for example, “Characterization of positive reproducing kernels. Application to Green’s functions,” by N. Aron-szajn and K. T. Smith [Am. J. Mathematics, vol. 79, pp. 611–622 (1957), http://www.jstor.org/stable/2372564].However, as usual there are pathological counter-examples.3There are many books with “nonnegative matrices” in their titles that cover this fact, usually as a special case of a moregeneral class of something called “M matrices,” but I haven’t yet found an elementary presentation at an 18.06 level. Notethat the diagonal entries of a positive-definite matrix P are always positive, thanks to the fact that Pii= eTiPei> 0 where eiisthe unit vector in