Residual Operator Mapping in SG-Calculus

• Residual operator mapping is a framework that extracts and utilizes invariant traces, like the Wodzicki residue, in SG-calculus.
• It employs double-order symbols and meromorphic zeta functions to define unique noncommutative traces on noncompact manifolds with cylindrical ends.
• The approach refines spectral asymptotics, supports a precise Weyl law, and bridges analysis with noncommutative geometry and mathematical physics.

Residual operator mapping denotes a set of mathematical and computational frameworks in which a “residual” is extracted, measured, or utilized as an invariant or fundamental quantity related to operators. This concept appears in analysis (noncommutative residues and pseudodifferential operators), geometry (trace functionals on noncompact manifolds), and spectral theory (Weyl asymptotics and spectral invariants). In the context of Wodzicki residue for operators on manifolds with cylindrical ends (Battisti et al., 2010), residual operator mapping identifies the unique trace in the algebra of SG-classical (double-order) pseudodifferential operators, canonically defined through the meromorphic continuation of the associated zeta function, and used for precise spectral asymptotics.

1. SG-Classical Operators and Double Order Structure

SG-classical operators, forming part of the so-called SG-calculus (symbolic calculus “with double order”), generalize standard pseudodifferential operators to noncompact domains such as ℝⁿ and manifolds with “cylindrical ends.” An operator $A$ is SG-classical of order $(m_1, m_2)$ if its symbol $a(x, \xi)$ satisfies: $$|\partial_x^\alpha \partial_\xi^\beta a(x, \xi)| \leq C_{\alpha\beta} \langle \xi \rangle^{m_1 - |\beta|} \langle x \rangle^{m_2 - |\alpha|},$$ for all multi-indices $(\alpha, \beta)$, ensuring precise control over regularity and decay in both Fourier ($\xi$) and spatial ($x$) variables. Classicality requires an asymptotic expansion in homogeneous components with respect to both variables. The SG-calculus supports spectral analysis and functional calculus on noncompact manifolds, bridging the gap between compact and noncompact cases.

2. Wodzicki Residue: Definition and Zeta Function Construction

The Wodzicki residue $TR(A)$, originally defined for classical pseudodifferential operators on compact manifolds, extends here to SG-classical operators. Its construction uses the associated zeta function of the operator’s complex power:

• Complex powers of $A$, $A^z$, defined via contour integrals in the complex plane (functional calculus: $$A^z = (1/2\pi i) \int_\Gamma \lambda^z (A - \lambda I)^{-1} d\lambda$$).
• The zeta function $$\displaystyle \left((A, z)\right) = Sp(A^z) = \int_{\mathbb{R}^n} K_{A^z}(x,x) dx$$, where $K_{A^z}(x,x)$ is the Schwartz kernel’s diagonal.

For sufficiently negative $\Re z$, $A^z$ is trace-class and so $\left((A, z)\right)$ is analytic. The crucial result is that $\left((A, z)\right)$ has a meromorphic extension to $\mathbb{C}$, with at most double poles. The Wodzicki residue is then the coefficient of the second-order pole at $z = 1$: $$TR(A) = m_1 m_2 \cdot \text{Res}^2_{z=1} \left((A, z)\right) = m_1 m_2 \cdot \lim_{z \to 1} (z-1)^2 \left((A, z)\right).$$ This trace is the unique invariant (up to multiples) on the algebra of SG-classical operators modulo smoothing operators. It isolates the “residual” or nonlocal part of the operator, functioning as a noncommutative trace.

3. Extension to Manifolds with Cylindrical Ends

The framework adapts SG-calculus and residual trace theory to manifolds with cylindrical ends, which combine a compact core and an asymptotically cylindrical noncompact “end” (parameters $p \in [T, \infty)$, angles $\omega \in S^{n-1}$). The SG structure is preserved by an admissible atlas, compatible partition of unity, and exit chart parameterizing the end.

• The definitions of symbols, complex powers, and zeta functions transfer locally via these charts.
• Technical challenges include invariance of homogeneous expansions under coordinate changes, and the global gluing of local traces ($((A,z))$ remains meromorphic globally).
• Under these constructions, the Wodzicki residue is well-defined globally for elliptic, SG-classical operators on cylindrical-end manifolds.

4. Ellipticity, Spectral Properties, and Functional Calculus

Successful application of the residual trace and zeta function requires elliptic operators with strong symbolic properties:

• Principal symbols invertible for large $|x| + |\xi|$ (“SG-elliptic” with respect to closed sectors $\Lambda$ in the complex plane).
• Complete asymptotic expansions with respect to $x$ and $\xi$.
• Discrete spectrum, invertibility on $L^2$, and kernel components compactly embedded via weighted Sobolev norms.

These conditions enable construction of complex powers, guarantee meromorphic continuation of the zeta function, and uniquely define the residual trace as a function of operator-symbol homogeneity.

5. Refined Weyl Law and Spectral Asymptotics

A key application is the derivation of a refined Weyl law for positive selfadjoint, elliptic SG-classical operators, particularly when $m_1 = m_2 = m$: $$N_A(X) = C_0 X^{1/m} \log X + C_1 X^{1/m} + \text{lower order terms},$$ where $N_A(X)$ is the eigenvalue counting function. In this formula,

• $C_0$ is proportional to $TR(A^{-1})$ after normalization.
• $C_1$ is an “angular term” arising from homogeneous expansions.
• The log-term is a unique consequence of the cylinder geometry and the residual trace, whereas the power-law term is present in both compact and cylindrical cases.

Explicit splitting of the trace is given by (see Theorem 2.4): $$TR_{I,\varepsilon}(A) = -TR_y(A) - TR_e(A) + \frac{1}{m_1 m_2} TR_\partial(A),$$ where $TR_y$ and $TR_e$ stem from separate homogeneous contributions, and $TR_\partial$ reflects the angular correction. This detailed expansion allows calculation of spectral invariants in geometrically singular or noncompact domains.

6. Implications in Analysis, Geometry, and Physics

The extension of the Wodzicki residue and SG-calculus to noncompact manifolds enables:

• Construction of noncommutative residue traces in the noncompact setting, generalizing the classical uniqueness properties.
• Determination of spectral invariants and correction terms in eigenvalue asymptotics, relevant for spectral geometry and global analysis.
• Bridges to noncommutative geometry, quantum field theory, and geometric models of gravity, where “residual” traces encode physical information (e.g., spectral actions, anomalies).
• A unified framework capable of handling spectral problems and index theory on manifolds with ends, with precise distributional control.

7. Summary Table: Key Properties of Residual Operator Mapping in SG-Calculus

Feature	SG-Classical Operators	Manifolds with Cylindrical Ends
Symbol Double Order	$(m_1, m_2)$, controls decay	Adapted via charts in noncompact regions
Zeta Function	$\left((A, z)\right)$ meromorphic	Meromorphic globally by construction
Residual Trace Formula	$$TR(A) = m_1 m_2 \cdot \lim_{z \to 1} (z-1)^2 ((A, z))$$	Same, with local-to-global gluing
Weyl Law Structure	2-term asymptotics (log and power)	Log-term unique to cylindrical geometry
Spectral Invariants	Computed directly via residue	Linked to noncommutative geometry & physics

The residual operator mapping, through the Wodzicki residue for SG-classical operators, provides a uniquely determined trace functional foundational for analysis and geometry on noncompact manifolds with cylindrical ends. This framework supports precise spectral invariants, refines classical asymptotics, and extends trace theory to global analysis and mathematical physics.