MATH 256B: PROPAGATION PHENOMENA We now understand elliptic operators in Ψm;`; the next challenge is to deal with non-elliptic operators. Let's start with classical operators, and indeed let's take 1 n n m = ` = 0. Thus, A = qL(a), a 2 C (R × R ), so σ0;0(A) is just the restriction n n of a to @( × ). Ellipticity is just the statement that a0 = aj n n does not R R @(R ×R ) vanish. Thus, the simplest (or least degenerate/complicated) way an operator can n n be non-elliptic is if a0 is real-valued, and has a non-degenerate 0. As @(R × R ) is not a smooth manifold at the corner, @Rn × @Rn, one has to be a bit careful. Away from the corner non-degeneract is the statement that a0(α) = 0 implies da0(α) 6= 0; −1 1 in this case the characteristic set, Char(A) = a0 (f0g) is a C codimension one embedded submanifold. At the corner, for α 2 @Rn × @Rn, one can consider the two smooth manifolds with boundary Rn × @Rn and @Rn × Rn, and denoting by a0;fiber and a0;base the corresponding restrictions, ask that a0(α) = 0 implies that da0;fiber and da0;base are not in the conormal bundle of the corner; in this case in both boundary hypersurfaces Char(A) is a smooth manifold transversal to the boundary. In many cases, such as stationary (elliptic) Schr¨odingeroperators, the characteristic set is disjoint from the corner, Char(A) is disjoint from the corner, but this is not the case for wave propagation. More generally, if A is classical, i.e. a = hzi`hζima~,a ~ 2 C1(Rn × Rn), we impose the analogous condition ona ~, i.e. thata ~0 has a non-degenerate zero set. Note 1 n n that if b 2 C (R × R ) is elliptic, anda ~0 has a non-degenerate zero set, then, with b0 denoting the restriction of b to the boundary, the same holds for b0a~0 as d(b0a~0) = b0 da~0 +a ~0 db0, so at when b0a~0 = 0, i.e. whena ~0 = 0, d(b0a~0) is a non-vanishing multiple of da~0. Let's re-interpret this from the conic point of view, for instance corresponding to Rn × @Rn, i.e. working on Rn × (Rn n f0g). (Note that working with @Rn × Rn, i.e. (Rn n f0g) × Rn is completely analogous.) Away from the corner, we may drop the compactification from the first factor, and thus we may assume ` = 0. For A m classical, then, σfiber;m;0(A) is homogeneous of degree m, given by am = jζj a~0, m wherea ~0 is considered as a homogeneous degree zero function. Now, d(jζj a~0) = m m jζj da~0 +a ~0 djζj , and Char(A) is given bya ~0 = 0, so it is now a conic (invariant under the R+-action) smooth codimension one embedded submanifold of Rn ×(Rn n f0g). The next relevant structure arises from the Hamilton vector field of A, n X Ham = (@ζj am)@zj − (@zj am)@ζj : j=1 Note that Ham am = 0, thus this vector field is tangent to Char(A), and thus defines a flow on Char(A). Note that Ham is homogeneous of degree m−1 in the sense that the push-forward of Ham under dilation in the fiber by t > 0, Mt(z; ζ) = (z; tζ), −m+1 is t Ham since, using that @zj am, resp. @ζj am are homogeneous of degree m, 1 2 resp. m − 1, n ∗ X 1−m (Ham Mt f)(z; ζ) = t (@ζj am)(z; tζ)(@zj f)(z; tζ) j=1 −m 1−m − t (@zj am)(z; tζ)t(@ζj f)(z; tζ) = t (Ham f)(z; tζ): In particular, integral curves of Ham through (z; ζ) and (z; tζ) are the `same' up to reparameterization: denoting the first by γ, the second byγ ~, and denoting by M~ t ~ dilations on R: Mt(s) = ts, m−1 m−1 ~ γ~(s) = (z(γ(t s)); tζ(γ(t s))) = (Mt ◦ γ ◦ Mtm−1 )(s); ~ d d since (Mt)∗ ds = t ds , so d m−1 d m−1 (M ◦ γ ◦ M~ m−1 ) = t (M ◦ γ) = t (M ) H = H ; t t ∗ ds t ∗ ds t ∗ am am d as being an integral curve of Ham means exactly that the push-forward of ds under the map is exactly Ham . In particular, up to reparameterization, the integral curves can be considered curves on Rn × (Rn n f0g)=R+, i.e. on this quotient, the image of the curve is defined, but not the parameterization itself. The exception is if m = 1, in which case even the parameterization is well-defined on this quotient; in this case, Ham acts on homogeneous degree zero functions, and is thus a vector field on the quotient. In general, if m is arbitrary, one may consider instead the vector field −m+1 jζj Ham , which is homogeneous of degree 0, and thus has well-defined integral curves on the quotient; this does depend on the choice of a positive homogeneous −m+1 degree 1 function, such as jζj, but a different choice only multiplies jζj Ham by a smooth non-vanishing function, and thus simply re-parameterizes the integral curves. A different way of looking at this is that for m = 1, by the identification of Rn × (Rn n f0g)=R+ with Rn × @Rn, one has a vector field on Rn × @Rn; if m 6= 1, then again this vector field is defined up to positive multiples. n n Definition 1. The integral curves of Ham in Char(A) ⊂ R × (R n f0g), as well as their projections to Rn × (Rn n f0g)=R+ = Rn × Sn−1 = Rn × @Rn, are called null-bicharacteristics, or simply bicharacteristics. The Hamiltonian version of classical mechanics corresponding to an energy func- tion p is just the statement that particles move on integral curves of Hp. In physical terms, geometric optics for the wave equation is that at high frequencies waves fol- low paths given by classical mechanics. Concretely, including the time variable as n n−1 part of our space, on Rz = Ry × Rt, one has a Lorentzian metric g, for instance a product-type metric, g = dt2 − h(y; dy), and then the dual metric function G on n n Rz × Rζ , given by G(z; ζ) = Gz(ζ; ζ), with Gz the inverse of gz. Note that G is homogeneous of degree 2. Then HG is a vector field of homogeneity degree 1. The integral curves of HG are the lifted geodesics; those inside fG = 0g are the lifted null-geodesics. Now, the d'Alembertian −1=2 X 1=2 g = j det gj Dij det gj GijDj has principal symbol G, while waves are solutions of gu = 0. Mathematically, the high frequency statement translates into singularities of solutions u. Concretely, we have the following theorem: 3 Theorem 0.1. Suppose that P 2 Ψm;0 and its principal symbol has a real homo- s s−m+1 geneous degree m representative p. Then in Rn × @Rn, WF (u) n WF (P u) is a union of maximally extended (null)-bicharacteristics of p. We now turn to the general case, where the characteristic set Char(A) of A 2 m;` Ψcl possibly intersects the corner. So first we define the rescaled Hamilton vector field −`+1 −m+1 Ha = Ha;m;` = hzi hζi Ha; n n and notice that as hzi@zj and hζi@ζj are in Vb(Rz ), resp. Vb(Rζ ), and thus in n n Vb(Rz × Rζ ), n n Ha;m;` 2 Vb(Rz × Rζ ): As above, we note that if we replace hzi−`+1hζi−m+1 by bhzi−`+1hζi−m+1, where 1 n n n n b 2 C (R × R ) is positive on @(R × R ), then Ha;m;` is only multiplied by n n a positive factor, b, at Char(A) ⊂ @(R × R ). Since Ha;m;` is tangent to the boundary, its integral curves are globally well-defined (i.e. they are well-defined on R). We then make the definition: n n Definition 2. The integral curves of Ha;m;` in Char(A) ⊂ @(R × R ), are called null-bicharacteristics, or simply bicharacteristics, and we consider them well-defined up to a direction-preserving reparameterization. In general, we have the following theorem, which contains Theorem 0.1 as a special case: m;` Theorem 0.2. Suppose that P 2 Ψcl , with real principal symbol p. Then in s;r s−m+1;r−`+1 @(Rn × Rn), WF (u) n WF (P u) is a union of maximally extended (null)-bicharacteristics of p. A variant of this theorem in the variable order setting is the following: m;` Theorem 0.3. Suppose that P 2 Ψcl , with real principal symbol p, and sup- pose s 2 C1(Rn × @Rn), r 2 C1(@Rn × Rn) are non-increasing along the rescaled n n s;r Hamilton flow, i.e. Hp;m;`s ≤ 0, Hp;m;`r ≤ 0. Then in @(R × R ), WF (u) n WFs−m+1;r−`+1(P u) is a union of maximally forward extended (null)-bicharacteristics of p. The analogous conclusion holds if s, r are non-decreasing and `forward' is re- placed by `backward'. Note that all these theorems have empty statements at points at which Hp;m;` vanishes (or is radial in the conic setting, i.e. is a multiple of the generator of dilations in the conic variable), so there is nothing to prove at such points.