Gaussian beams concentrated on an embedded null bicharacteristic

Write $\beta =\pi \circ \mathrm{\Gamma }$ for the base curve. Cover its parameter interval by finitely many small overlapping intervals on which $\beta$ is embedded. In an ambient coordinate chart choose a coordinate $x_1$ whose derivative along $\beta$ is nonzero, and shrink the interval so $x_1\circ \beta$ is strictly monotone. Its inverse is continuous by monotonicity; the difference quotient gives derivative $1\mathrm{/}(x_1\circ \beta {)}^{\mathrm{\prime }}$, and repeated differentiation gives smooth higher derivatives. The coordinate change $s=(x_1\circ \beta {)}^{-1}(x_1)$ and $y_j=x_j-x_j(\beta (s))$ has a triangular smooth inverse, so the branch is $y=0$ with its prescribed parameter $s$. Write $\xi (s)=({\xi }_s(s),{\xi }_y(s))$ in these coordinates. Prescribe ${\mathrm{\partial }}_y\varphi (s,0)={\xi }_y(s)$ and ${\mathrm{\partial }}_s\varphi (s,0)={\xi }_s(s)$, integrating the latter from a chosen real initial phase value. These first jets are compatible with the eikonal equation because $p(s,0,{\xi }_s(s),{\xi }_y(s))=0$. The central covector has $p_{{\xi }_s}\mathrm{\ne }0$ on a sufficiently small interval, since the Hamilton velocity in these coordinates is a nonzero multiple of ${\mathrm{\partial }}_s$. Expand the formal equation $p(s,y,{\mathrm{\partial }}_s\varphi ,{\mathrm{\partial }}_y\varphi )=0$ in transverse powers of $y$. At each degree the unknown coefficient satisfies a linear ODE in $s$ with nonzero leading coefficient $p_{{\xi }_s}$, so it is uniquely determined from its initial value; the coefficients depend smoothly on $s$. Its zeroth and first jet equations are exactly the given Hamilton curve (after its parameter change). Initialize the transverse Hessian by $H(0)=iI$ in the chosen transverse coordinates.

The quadratic jet equation is the matrix Riccati equation $H^{\mathrm{\prime }}+D+BH+H{B}^T+HCH=0,$ where $C,D$ are real symmetric and $B$ real, the corresponding blocks of the reduced Hamiltonian Hessian. Solve instead $Y^{\mathrm{\prime }}=B^{T}Y+CZ$, $Z^{\mathrm{\prime }}=-DY-BZ$, $Y(0)=I$, $Z(0)=iI$. Direct differentiation gives $Y^{T}Z-Z^{T}Y=0$ and $Y^{\ast }Z-Z^{\ast }Y=2iI$. If $Yv=0$ the second identity gives $0=2i\mathrm{\mid }v{\mathrm{\mid }}^2$, so $Y$ is invertible on the whole interval. Consequently $H=Z{Y}^{-1}$ is symmetric, solves the Riccati equation, and $\mathrm{Im}H=(Y^{-1}{)}^{\ast }{Y}^{-1}>0.$ Compactness gives uniform positive lower and finite upper bounds. At an overlap transform all jets by the coordinate chain rule, carry these initial data to the next chart, and use ODE uniqueness. Positivity is unchanged by real coordinate change and persists on the finite cover.

For transverse degree $k\ge 3$, differentiating the reduced eikonal equation $k$ times gives a finite linear ODE for the degree-$k$ phase jet: its highest jet enters linearly, with coefficients determined by the Hessian and the first derivatives of the reduced Hamiltonian; the inhomogeneous terms involve only lower jets and derivatives of the symbol. This follows by the product rule: terms containing a highest derivative of $\varphi$ can have only one such factor, and the coefficient of the transverse degree-$(k+1)$ jet is the transverse characteristic velocity $dy\mathrm{/}ds$, which is zero on the central curve $y=0$. Choose zero initial transverse coefficients for degrees $k\ge 3$ in the first chart, then solve successively for every degree using [F2]. On an overlap, transform the full jet by the coordinate chain rule before using ODE uniqueness, so the local jets agree. Apply [F1], or its explicit shrinking-cutoff series, to realize them. This yields a smooth phase with eikonal defect flat on each branch and $\mathrm{Im}\varphi \ge c\mathrm{\mid }y{\mathrm{\mid }}^2$ after shrinking the transverse tube.

Define $T_{\varphi }$ to be the coefficient of $h^{-1}$ in the exact conjugation identity $P(e^{i\varphi \mathrm{/}h}a)=e^{i\varphi \mathrm{/}h}(h^{-2}p(x,d\varphi )a+h^{-1}{T}_{\varphi }a+Pa)\mathrm{.}$ It is a first-order operator with derivative part $p_{\xi }(x,d\varphi )\cdot D$, so its coefficient of ${\mathrm{\partial }}_s$ on the ray is nonzero. Solve $T_{\varphi }{a}_0=0$ and $T_{\varphi }{a}_j=-P{a}_{j-1}$ for $j\ge 1$ to infinite transverse order. The central $a_0$ equation is a scalar homogeneous ODE with nonzero initial value, hence has nonvanishing exponential solution. For each successive transverse degree and amplitude index the differentiated equations are linear ODEs with already known right sides, by the same cancellation of derivatives along the curve as in step 3.1. Realize these jets by [F1]. The separate asymptotic series $a(x;h)\sim {\sum }_{j\ge 0}{h}^j{a}_j(x)$ is realized by ${\sum }_j{h}^j{a}_j(x)\eta (h\mathrm{/}{\delta }_j)$, where $\eta \in C_c^{\mathrm{\infty }}([0,\mathrm{\infty }))$ is one near zero and supported in $[0,1]$, and ${\delta }_j\downarrow 0$ is chosen so its $C^k$ tails are summable for every fixed $k$ and have the required $h$ asymptotic bounds. Choose $0<{\delta }_0<1$ freely. For each $j\ge 1$, choose $0<{\delta }_j<\min ({\delta }_{j-1},2^{-j})$ with ${\delta }_j^{j\mathrm{/}2}(1+\mathrm{\parallel }{a}_j{\mathrm{\parallel }}_{C^j})<2^{-j}$; this is possible because $j\mathrm{/}2>0$. For fixed $k,J$, each term with $j\ge \max (k,2J+2)$ has $C^k$ norm at most $\mathrm{\parallel }\eta {\mathrm{\parallel }}_{\mathrm{\infty }}{2}^{-j}{h}^J$, since its support has $h\le {\delta }_j<1$ and $j-J\ge j\mathrm{/}2$. Summing these bounds controls the tail. Each of the finitely many earlier cutoffs equals one for sufficiently small $h$, which proves the claimed asymptotic expansion, including its leading coefficient $a_0$. This proves the full transport and Borel realization.

On an embedded branch take $u_h=h^{-d\mathrm{/}4}{e}^{i\varphi \mathrm{/}h}a(x;h)$ and a fixed transverse cutoff. For a flat defect $r$, Taylor's formula gives $\mathrm{\mid }{D}^{\alpha }r\mathrm{\mid }\le C_{k,\alpha }\mathrm{\mid }y{\mathrm{\mid }}^k$ for every $k$. Scaling $y=\sqrt{h}z$ and using $\mathrm{Im}\varphi \ge c\mathrm{\mid }y{\mathrm{\mid }}^2$ shows $\mathrm{\parallel }{h}^{-d\mathrm{/}4}\mathrm{\mid }y{\mathrm{\mid }}^k{e}^{-c\mathrm{\mid }y{\mathrm{\mid }}^2\mathrm{/}h}{\mathrm{\parallel }}_{L^2}\le C_k{h}^{k\mathrm{/}2}\mathrm{.}$ Every application of $P$ loses at most two powers of $h$. Choose $k$ and the amplitude order large enough to dominate that loss and any requested $N$. Derivatives of the transverse cutoff are supported at fixed positive distance from the ray and are $O(h^{\mathrm{\infty }})$ by Gaussian decay. Thus the residual is $O(h^N)$ away from longitudinal endpoint cutoffs; at endpoints the commutator with a longitudinal cutoff remains, and no vanishing assertion is made there. This also proves uniform $L^2$ boundedness on each branch.

On each compact parameter interval choose finitely many smooth bump functions subordinate to the branch cover, positive on a smaller cover, and divide each by their sum. Such bumps follow from $\vartheta (t)=e^{-1\mathrm{/}t}$ for $t>0$ and $0$ otherwise, by the same nested-interval formula used for a compactly supported cutoff. Patch using the compatible phase and amplitude jets constructed above. On overlaps of descriptions of the same branch, their phases and amplitudes differ by flat terms, so after multiplication by $h^{-d\mathrm{/}4}{e}^{i\varphi \mathrm{/}h}$ their difference and any fixed number of derivatives are $O(h^N)$ for every $N$, by step 5.1. Derivatives of the partition cancel because its sum is one; the remaining flat differences have the same bound. For distinct parameter branches whose base tubes overlap, phase-space injectivity makes their central covectors distinct at every common base point. Shrink the finitely many overlap neighborhoods so that the gradient of the difference of their real phases is bounded away from zero. With $\theta =\mathrm{Re}({\varphi }_1-{\varphi }_2)$, choose a smooth vector field $V$ there with $V\theta =1$ and integrate by parts in $e^{i\theta \mathrm{/}h}$. Factoring out $e^{i\theta \mathrm{/}h}$, the Gaussian scaling in step 5.1 gives an $O(1)$ normalized $L^1$ bound for the product amplitude; each transverse derivative costs at most $h^{-1\mathrm{/}2}$ because $h^{-1}\mathrm{\mid }y\mathrm{\mid }=O(h^{-1\mathrm{/}2})$ on $\mathrm{\mid }y\mathrm{\mid }=O(\sqrt{h})$. Thus each integration contributes $h$ and costs at most $h^{-1\mathrm{/}2}$, so after $k$ integrations the mixed term is $O(h^{k\mathrm{/}2})$. Thus every mixed product tends to zero to arbitrary order. Away from overlaps ordinary Gaussian decay applies. Hence patching preserves the residual, positivity, and the sum of the branch mass limits, including at projected self-intersections.

Write the density in a branch as $\rho (s,y)\mathrm{\mid }dsdy\mathrm{\mid }$. For a smooth test function, set $y=\sqrt{h}z$. On each bounded $z$-set the rescaled integrand converges uniformly to the test value times $\mathrm{\mid }{a}_0(s,0){\mathrm{\mid }}^2\rho (s,0)e^{-z^T\mathrm{Im}H(s)z}$. Since $\mathrm{Im}\varphi \ge c\mathrm{\mid }y{\mathrm{\mid }}^2$ and the amplitudes are uniformly bounded, its absolute value is bounded by $C{e}^{-2c\mathrm{\mid }z{\mathrm{\mid }}^2}$; the integral of this bound outside $\mathrm{\mid }z\mathrm{\mid }\le R$ tends to zero as $R\to \mathrm{\infty }$, uniformly in $h$ and $s$. First integrate on $\mathrm{\mid }z\mathrm{\mid }\le R$, then let $R\to \mathrm{\infty }$. The Gaussian integral gives $\mathrm{\mid }{u}_h{\mathrm{\mid }}^{2}d{V}_X⟶w(s)ds,w(s)=\frac{{\pi }^{d\mathrm{/}2}\mathrm{\mid }{a}_0(s,0){\mathrm{\mid }}^2\rho (s,0)}{\sqrt{\det \mathrm{Im}H(s)}}\mathrm{.}$ This positive smooth function is fixed by the leading transport equation; choose the initial scalar amplitude to make $w(0)=1$. Coordinate changes preserve this expression as a density, and step 6.1 removes mixed terms. For a formally self-adjoint operator the leading transport equation and the Riccati determinant identity give $(p_{{\xi }_s}w{)}^{\mathrm{\prime }}=0$: this follows by multiplying $T_{\varphi }{a}_0=0$ by ${\bar{a}}_0$, subtracting its conjugate, and using ${\rho }^{-1}{\mathrm{\partial }}_j(\rho p_{{\xi }_j})$ for the real divergence term. For the real wave operator in physical-time coordinates the normalized central covector has $\mathrm{\mid }{\xi }_t\mathrm{\mid }=1$, hence $\mathrm{\mid }{p}_{{\xi }_t}\mathrm{\mid }=2$ is constant; its mass density is therefore constant and the chosen normalization gives $w=1$. Arbitrary complex lower-order terms instead enter the scalar amplitude ODE, giving precisely the transported weight above.

If the endpoint cutoff regions lie outside the measurement cylinder, step 5.1 gives $\mathrm{\parallel }P{u}_h{\mathrm{\parallel }}_{L^2(Q)}=O(h^N)$ for every $N$. In the energy-correction specialization with the exact cylinder, density and operator specified in the Statement, assume also the approximate beam vanishes near the initial slice for a forward correction or terminal slice for a backward correction. Apply [F3] to $-P{u}_h$ with zero boundary and the corresponding zero endpoint data. Its energy norm is $O(h^N)$, since $\mathrm{\parallel }P{u}_h{\mathrm{\parallel }}_{L_t^1{L}_x^2}\le \sqrt{T}\mathrm{\parallel }P{u}_h{\mathrm{\parallel }}_{L^2(Q)}$. Adding this correction gives an exact solution with the actual approximate boundary data and the same normalized wave ray limit; the product error is bounded by Cauchy–Schwarz and uniform $L^2$ bounds. This proves every claimed localization and correction assertion.