Parameter inverses and slowly quantized quadratics

A quadratic operator can be invertible for every fixed parameter without an obvious common inverse bound. Positive real energy controls large parameters; continuity controls the remaining compact region. Keeping both parts of the argument gives derivative bounds strong enough to quantize the parameter itself.

We use the maximal domain and graph estimate from The spectrum of a complex quadratic polynomial, quadratic positivity from Quadratic energy and the positive trace, and the Hilbert-coefficient operator bound in Section 7 of When a moving symbol scale controls an operator. That last bound allows infinite-dimensional coefficient spaces. The bounded inverse theorem is in Banach estimates, quotient spaces and compact parameter arguments.

Basic references are Nonnenmacher's open semiclassical lectures [N], Lerner's phase-space chapter [L], and Sjöstrand's multiple-characteristic paper [S]. The inverse differentiation formula is elementary. The semiclassical parametrix method combines Weyl ordering with the Calderón–Vaillancourt bound; here we supply the parameter weights and the exact quadratic remainder.

1. A parameter estimate with every derivative

Let $q(z,p)$ be a complex homogeneous quadratic form, with $z=(x,\xi)\in\mathbb R^{2m}$ and $p\in\mathbb R^r$, and assume $\operatorname{Re}q(z,p)\geq a(|z|^2+|p|^2),\qquad a>0. \tag{1.1}$ Fix $\kappa\in\mathbb C$, and set $P(p)=q^w(x,D,p)+\kappa,\qquad \rho(p)=(1+|p|^2)^{1/2}. \tag{1.2}$ Weyl quantization acts only in $z$. The parameter is real multiplication, not an additional momentum in (1.2).

Theorem 1.1. Suppose every $P(p):\mathcal B_2\to L^2$ is invertible. Its inverse $E(p)$ is smooth in operator norm $L^2\to\mathcal B_2$, is locally holomorphic in complex $p$, and satisfies, for every multi-index $\gamma$ and every word $Z$ of degree $j\leq2$, $\|Z\partial_p^\gamma E(p)\|_{L^2\to L^2} \leq C_{\gamma,Z}\rho(p)^{j-2-|\gamma|}. \tag{1.3}$ If $q_\nu\to q$ and $\kappa_\nu\to\kappa$ in their finitely many coefficients, the perturbed inverses exist for all real $p$ when $\nu$ is large, and (1.3) holds with constants independent of $\nu$. In particular no compactness restriction on the real parameter is imposed.

Proof. The quadratic graph estimate gives $P(p):\mathcal B_2\to L^2$ bounded with this exact maximal domain. Its finitely many coefficients depend polynomially on $p$. Inversion near any real $p_0$ follows from $P(p)=\bigl(I+(P(p)-P(p_0))E(p_0)\bigr)P(p_0). \tag{1.4}$ The geometric inverse series converges when the perturbation norm is less than one. It gives smoothness and holomorphic dependence on a complex neighborhood of $p_0$, in the claimed spaces. A finite cover bounds $E$ into $\mathcal B_2$ on each compact parameter set.

For large $|p|$, quantize the nonnegative real quadratic form $\operatorname{Re}q-a(|z|^2+|p|^2)$. The exact real quadratic positivity theorem gives, for $u\in\mathcal S$, $\operatorname{Re}(P(p)u,u) \geq a\sum_{j=1}^m(\|x_ju\|^2+\|D_ju\|^2) +(a|p|^2+\operatorname{Re}\kappa)\|u\|^2. \tag{1.5}$ Thus, outside a fixed ball, with $f=P(p)u$, $\|u\|\leq C\rho^{-2}\|f\|,\qquad \sum_j(\|x_ju\|+\|D_ju\|) \leq C\rho^{-1}\|f\|. \tag{1.6}$ For the first inequality use Cauchy–Schwarz in (1.5) and absorb the fixed $\operatorname{Re}\kappa$. Substitution of that first inequality into (1.5) proves the second.

The graph estimate at $p=0$ says $\|u\|_{\mathcal B_2}\leq C(\|P(0)u\|+\|u\|). \tag{1.7}$ The difference $P(p)-P(0)$ is a linear combination of $|p|$ times first words and $|p|^2$ times the identity, with fixed bounded coefficients. Equations (1.6) therefore bound its value on $u$ by $C\|f\|$. Equation (1.7) gives all second-word bounds. Together with compact-parameter inversion, this proves (1.3) for $\gamma=0$. Smooth compact approximation in $\mathcal B_2$ extends the estimates to the full domain, so they apply to $u=E(p)f$.

Differentiate $P(p)E(p)=I$: $\partial_{p_j}E=-E(\partial_{p_j}P)E. \tag{1.8}$ A derivative of $P$ of order $b=1$ has coefficients of size $C\rho$ on the identity and bounded coefficients on first words. A derivative of order $b=2$ is a scalar constant operator. Higher derivatives vanish. Hence $\|(\partial_p^\eta P)E\|\leq C_\eta\rho^{-|\eta|}, \qquad 1\leq|\eta|\leq2. \tag{1.9}$ Repeated differentiation of (1.8) gives a finite sum of ordered products $E(\partial^{\eta_1}P)E\cdots (\partial^{\eta_s}P)E,\qquad |\eta_1|+\cdots+|\eta_s|=|\gamma|,\quad 1\leq|\eta_j|\leq2. \tag{1.10}$ Put $Z$ on the first inverse. Its bound is $C\rho^{j-2}$; every subsequent factor pair obeys (1.9). This proves (1.3), with no commutation of these factors.

For perturbations write $\epsilon_\nu$ for the maximum coefficient difference, including $\kappa$. Polynomial degree and the zeroth-derivative bounds give $\sup_{p\in\mathbb R^r} \|(P_\nu(p)-P(p))E(p)\|\leq C\epsilon_\nu. \tag{1.11}$ Indeed each term is a coefficient difference times a second word, $|p|$ times a first word, or $(1+|p|^2)$ times the identity, applied to $E$. When $C\epsilon_\nu<1/2$, $E_\nu=E\bigl(I+(P_\nu-P)E\bigr)^{-1} \tag{1.12}$ has the same weighted zeroth-derivative bounds. The polynomial coefficients of $P_\nu$ remain bounded. Applying the ordered differentiation argument to $E_\nu$ proves every remaining uniform bound.

The same calculation for a complex increment $h$ gives $\|(P(p+h)-P(p))E(p)\| \leq C\bigl(|h|/\rho+|h|^2/\rho^2\bigr). \tag{1.13}$ Thus the holomorphic neighborhoods can include balls of radius $c\rho(p)$, for a fixed small $c>0$; the perturbed families have the same property. This proves the theorem. ∎

When $m=0$, the coefficient Hilbert space is $\mathbb C$, and $\mathcal B_2=\mathbb C$. The proof reduces to the inverse of the nonzero scalar $q(p)+\kappa$, including the compact/large-parameter division and all derivatives. When $r=0$, only the unparameterized inverse is asserted.

2. An operator-valued Weyl estimate with several small scales

Split the real parameter as $p=(Y,\Xi,t)$, with $Y,\Xi\in\mathbb R^d$ and $t\in\mathbb R^\ell$. The variable $t$ remains multiplication. For $0<h_j\leq1$, define the Weyl operator in $Y$ by $\begin{split} (\operatorname{Op}_{h}^{w}(A)u)(Y) ={}&(2\pi)^{-d}\prod_jh_j^{-1} \iint e^{i\sum_j(Y_j-V_j)\Xi_j/h_j}\\ &\hspace{12mm}\cdot A((Y+V)/2,\Xi,t)u(V)\,dV\,d\Xi. \end{split} \tag{2.1}$ Here $A$ takes values in bounded maps between Hilbert spaces. Oscillatory integrals have their distributional meaning, and boundedness follows from the linked Hilbert-coefficient theorem.

Specifically there are an integer $N$ and a constant $C_d$, independent of all the $h_j$, such that $\|\operatorname{Op}_{h}^{w}(A)\| \leq C_d\max_{|\gamma|\leq N} \|\partial_{Y,\Xi}^{\gamma}A\|_\infty. \tag{2.2}$ To check the specialization, the unitary dilation $Y_j=\sqrt{h_j}\,y_j$ turns (2.1) into ordinary Weyl quantization of $A(\sqrt h\,y,\sqrt h\,\eta,t)$. Each derivative acquires a factor at most one. Apply the coefficient theorem with the constant Euclidean metric, whose dual is itself, and with weight one. No ratio $h_j/h_k$ needs a bound. The same estimate holds between different coefficient Hilbert spaces. Uniformity in $t$ permits integration in that variable.

3. Quantizing a family of inverses

Assume the hypotheses of Theorem 1.1 for $q(z,Y,\Xi,t)$. Write $P(Y,\Xi,t)=q^w(x,D_x,Y,\Xi,t)+\kappa. \tag{3.1}$ Quantize the remaining parameter pair in (3.1): $\mathcal P_h=\operatorname{Op}_{h}^{w}(P),\qquad B_h=\operatorname{Op}_{h}^{w}(E). \tag{3.2}$ Quantization of $P$ is a quadratic differential operator in $x,Y$, with $h_jD_{Y_j}$ in place of $\Xi_j$; Weyl ordering is used in both sets of variables. We do not treat $P$ as a bounded $L^2_x$-valued symbol: it acts from $\mathcal B_2$ to $L^2_x$.

Proposition 3.1. There are $h_0>0$ and $C<\infty$ such that $\|u\|_{L^2_{x,Y,t}}\leq C\|\mathcal P_hu\|_{L^2_{x,Y,t}}, \qquad u\in\mathcal S,\quad \max_jh_j\leq h_0. \tag{3.3}$ The constants also work for the sufficiently small coefficient perturbations in Theorem 1.1.

Proof. Equation (1.3) and (2.2) bound $B_h$ uniformly on $L^2$. They also give all the domain control needed to apply $\mathcal P_h$. If a word in $x,D_x$ has degree $j$, its action on $E$ has size $C\rho^{j-2}$. Multiplication by a parameter monomial of degree $b$, for $b+j\leq2$, keeps all of its parameter derivatives bounded. Weyl composition with $Y$ or $hD_Y$ adds only finitely many parameter derivatives of $E$, with bounded factors $h_j$. Applying (2.2) shows that every total weighted word of degree at most two in $x,Y,D_x,hD_Y$, as well as $t^2$, is bounded after $B_h$. Mixed first words are included. Thus $\mathcal P_hB_h$ is defined distributionally and maps $L^2$ to $L^2$.

The exact finite Weyl product is $P\#_h E=I+R_h, \tag{3.4}$ where $\begin{split} R_h={}&\frac1{2i}\sum_jh_j (P_{\Xi_j}E_{Y_j}-P_{Y_j}E_{\Xi_j})\\ &-\frac18\sum_{j,k}h_jh_k (P_{\Xi_j\Xi_k}E_{Y_jY_k} -2P_{\Xi_jY_k}E_{Y_j\Xi_k} +P_{Y_jY_k}E_{\Xi_j\Xi_k}). \end{split} \tag{3.5}$ The order of multiplication is the one displayed. All derivatives of $P$ of parameter order at least three vanish.

For precision, this finite identity follows by applying $Y_j$ and $h_jD_{Y_j}$ to the Weyl kernel (2.1). Split each multiplication into its midpoint part and the difference from the midpoint, then integrate the difference by parts in $\Xi_j$. Differentiation of the midpoint contributes its factor $1/2$. Repeating twice gives (3.5), including its sign and coefficient. Operators in $x$ act on the coefficient $E$ on their indicated side. Pair first with Schwartz tests in $x,Y$, insert compact phase cutoffs, and integrate by parts until the scalar pairings are absolutely integrable. The derivative bounds in (1.3) and the bounded word combinations just established allow removal of these cutoffs in distributional pairings. This proves $\mathcal P_hB_h=I+\operatorname{Op}_h^w(R_h)$. It uses a finite differential identity, rather than a bounded-symbol composition assertion for the unbounded coefficient $P$.

A parameter derivative of $P$ of order $b$ has terms $C\,p^\delta Z,\qquad |\delta|+\deg Z\leq2-b. \tag{3.6}$ Multiply such a term by a parameter derivative of $E$ of order $c$. Equation (1.3) bounds the result by $C\rho^{-b-c}$. The differentiated versions have an additional inverse power for each further derivative. Every term of the first line of (3.5) consequently has size $C(\max h_j)\rho^{-2}$, and every second-line term has size $C(\max h_j)^2\rho^{-4}$. Thus (2.2) gives $\|\operatorname{Op}_h^w(R_h)\|\leq C\max_jh_j. \tag{3.7}$

We need a left identity to prove a lower norm estimate. Apply the same right-identity argument to the formal adjoint fiber $P(p)^*$. Its inverse is $E(p)^*$, with the same type of bounds by Theorem 1.1 applied to $\overline q,\overline\kappa$. The quadratic maximal domains justify this adjoint assertion. Adjointing its right identity gives $B_h\mathcal P_h=I+\widetilde R_h,\qquad \|\widetilde R_h\|\leq C\max_jh_j \tag{3.8}$ on Schwartz functions; the Weyl kernel gives $B_h^*=\operatorname{Op}_h^w(E^*)$. Choose $h_0$ so that the last norm is at most $1/2$. From (3.8), $\tfrac12\|u\|\leq\|B_h\|\|\mathcal P_hu\|,$ which proves (3.3). Every seminorm used above is uniform under the coefficient perturbations of Theorem 1.1. ∎

For $d=0$, (3.3) is the uniform fiber estimate, squared and integrated in $t$. No small parameter or Weyl remainder occurs.

4. Examples and exercises with complete solutions

Exercise 1 — the parameter weight, 6 points. With no $x$ variable, take $P(p)=1+|p|^2$. Verify the zeroth- and first-derivative estimates of Theorem 1.1. Explain what changes for $P(p)=p^2-1$ on the real line.

Solution. The inverse is $\rho^{-2}$, and $\partial_{p_j}E=-2p_j\rho^{-4}$, whose absolute value is at most $2\rho^{-3}$. Repeated differentiation gives sums of polynomials divided by higher powers of $1+|p|^2$, with bound $C_\gamma\rho^{-2-|\gamma|}$; this also follows from (1.10). The second polynomial is zero at $p=\pm1$, so lacks real fiber inverses. Large-parameter growth alone does not repair those two failures.

Exercise 2 — a perturbation growing with the parameter, 8 points. Let $P_0(p)=D_x^2+x^2+1+p^2$, and perturb it by $i\epsilon p x$. Show that $(P_\epsilon-P_0)E_0$ has norm at most $C|\epsilon|$, uniformly for real $p$.

Solution. The perturbation is $i\epsilon p x$. Theorem 1.1 gives $\|xE_0(p)\|\leq C\rho^{-1}$; hence

\|(P_\epsilon-P_0)E_0(p)\| \leq C|\epsilon|\frac{|p|}{\rho} \leq C|\epsilon|.

The geometric series makes every real perturbed fiber invertible for small $|\epsilon|$. The unweighted coefficient $\epsilon p$ is not uniformly small, but its action after the weighted inverse is.

Exercise 3 — an exact scalar remainder, 10 points. In one parameter pair put $P(Y,\Xi)=1+Y^2+\Xi^2$ and $E=P^{-1}$. Compute $P\#_hE-1$. Why is the scalar reciprocal not already the exact operator inverse?

Solution. The first bracket in (3.5) vanishes because $E$ is a function of $P$. The second bracket is $2E_{YY}+2E_{\Xi\Xi}$. Thus the difference is $-h^2\Delta E/4$. With $r^2=Y^2+\Xi^2$, direct differentiation gives $\Delta E=4(r^2-1)/(1+r^2)^3$, so $P\#_hE=1+h^2\frac{1-r^2}{(1+r^2)^3}.$ The extra bounded symbol is nonzero. Weyl quantization respects the product through its exact Weyl correction, not through pointwise multiplication. In this example the leading remainder is of order $h^2$, better than the general order-$h$ bound.

Exercise 4 — unequal rates, 6 points. Suppose $h_1=\epsilon$, $h_2=\epsilon^3$, with $\epsilon\to0$. Which small quantity controls (3.7)? Does the proof require comparable rates?

Solution. The bound is $C\max(h_1,h_2)=C\epsilon$. The dilation proving (2.2) uses each square root separately, and every product correction carries its own $h_j$ or $h_jh_k$. Both scales are at most one and tend to zero. Their ratio need not stay bounded away from zero.

Exercise 5 — why the left identity matters, 8 points. On $\ell^2(\mathbb N)$ let $S(a_0,a_1,\ldots)=(0,a_0,a_1,\ldots)$. Show that $S^*S=I$, but $S^*$ has no lower norm bound. Relate this to Proposition 3.1.

Solution. The backward shift $S^*$ removes the first coordinate. Hence $S^*S=I$, while $S^*e_0=0$. The identity is a bounded right inverse for $S^*$, and says nothing by itself about this kernel. Here $SS^*=I-\langle\,\cdot,e_0\rangle e_0$, whose remainder has norm one. Proposition 3.1 establishes a left remainder of norm less than one half, permitting absorption and excluding exactly this obstruction.

References

[N] Stéphane Nonnenmacher, An Introduction to Semiclassical Analysis, lectures, September–November 2020. Open university-hosted notes. Theorem 4.26 gives the semiclassical Calderón–Vaillancourt bound; Section 5 explains the parametrix method.
[L] Nicolas Lerner, Metrics on the Phase Space, chapter 2 of Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators. Author-hosted chapter.
[S] Johannes Sjöstrand, Parametrices for pseudodifferential operators with multiple characteristics, Arkiv för Matematik 12 (1974), 85–130. Original paper.

Written by GPT-6.1 Sol (OpenAI), at Ultra reasoning effort, September 2026. Self-checked by the writing AI. Public domain (CC0).