DRSN X: Mathematical Foundations of Drift Geometry (De Rerum Spectrale Natura, Report X, Version 1.0)

DSRN X: Mathematical Foundations of Drift Geometry De Rerum Spectrale Natura series REPORT X (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 •Rigorous functional–analytic formulation of drift geometry. •Unbounded operators, domains, and similarity flows. •Kato theory and holomorphic families of Dirac operators. •Spectral invariants under drift deformation. •Mathematical closure of the DSRN programme. DSRN X: Mathematical Foundations of Drift Geometry: Functional–Analytic Structure of Drifted Dirac Operators J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We present a rigorous mathematical formulation of drift geometry, focusing on the functional– analytic foundations underlying drifted Dirac operators. The drift deformation is analysed as a similarity flow on unbounded self–adjoint operators, preserving spectral properties while modifying lower–order structure. We establish domain stability, self–adjointness, spectral invariance, and holomorphic dependence using Kato theory. This report provides the mathematical closure of the DSRN programme, isolating the operator–theoretic core underlying its geometric, physical, and quantum applications. Keywords: Unbounded Operators; Dirac Operators; Spectral Theory; Kato Theory; Holomorphic Families; Noncommutative Geometry; Drift Geometry. CONTENTS I. Introduction 3 II. Functional–Analytic Framework of Drift Geometry 4 A. Hilbert Space Setting 4 B. Drift as a Similarity Flow 4 C. Basic Structural Properties 4 III. Holomorphic Families, Kato Theory, and Analytic Drift 5 A. Kato Type-(A) Families 5 B. Analytic Resolvents and Functional Calculus 6 C. Spectral Projections and Analytic Continuation 6 D. Generator of the Drift Flow 6 E. Consequences for Spectral Geometry 7 IV. Pseudodifferential Order, BCH Expansion, and Drift Corrections 7 ∗jp[email protected] 3 A. Pseudodifferential Operators and Order 7 B. BCH Expansion of the Drifted Operator 7 C. Stability of the Dirac Class 8 D. Drift Corrections to the Square 8 E. Consequences for Heat Kernel Asymptotics 8 F. Summary 9 V. Heat Kernel, Zeta Functions, and Spectral Invariants under Drift 9 A. Heat Kernel under Similarity Transformations 9 B. Heat Kernel Asymptotics 10 C. Spectral Zeta Functions 10 D. Eta Invariants and Spectral Asymmetry 10 E. Spectral Action Invariance 11 F. Consequences for Geometry and Physics 11 VI. Closure of Drift Geometry, Open Problems, and Outlook 12 A. Mathematical Closure of Drift Geometry 12 B. Conceptual Significance 12 C. Open Mathematical Problems 13 D. Relation to the DSRN Programme 13 E. Outlook 14 A. Summary of Core Results 14 References 14 I. INTRODUCTION The concept of drift geometry introduced in the preceding reports rests on a simple but subtle operator–theoretic idea: a similarity deformation of a Dirac operator by a bounded generator. Despite its apparent simplicity, this construction raises delicate mathematical questions concerning domains, self–adjointness, spectral invariants, and analytic dependence. The purpose of the present report is to provide a rigorous and self–contained functional–analytic foundation for drift geometry. All physical interpretations are deliberately set aside. Instead, we 4 focus exclusively on unbounded operators on Hilbert spaces and their behaviour under similarity flows. By isolating the mathematical core of drift geometry, this report serves both as a validation of the constructions used throughout the DSRN programme and as an independent contribution to operator theory and spectral geometry. II. FUNCTIONAL–ANALYTIC FRAMEWORK OF DRIFT GEOMETRY A. Hilbert Space Setting Let Hbe a complex separable Hilbert space. We consider densely defined, closed operators D: Dom(D)⊂ H → H that are self–adjoint and admit compact resolvent. Such operators include geometric Dirac operators on compact manifolds and internal Dirac operators of finite spectral triples. B. Drift as a Similarity Flow Let φbe a bounded self–adjoint operator on H. For each s∈R, define Ds:= esφDe−sφ.(II.1) Since esφ is bounded and invertible, Dsis well defined on Dom(Ds)=esφ Dom(D). Definition 1 (Drift Flow).The family {Ds}s∈Ris called the drift flow generated by φ. C. Basic Structural Properties We now collect the fundamental analytic properties of the drift flow. Theorem 2 (Domain Stability).For all s∈R , the operator Ds is densely defined and closed, with Dom(Ds)=esφ Dom(D). Theorem 3 (Self–Adjointness).If D is self–adjoint and φ is bounded and self–adjoint, then Ds is self–adjoint for all s∈R. 5 Theorem 4 (Spectral Invariance).For all s∈R, σ(Ds)=σ(D), and the resolvents satisfy (Ds−z)−1=esφ(D−z)−1e−sφ. These results establish drift geometry as a similarity–based deformation that preserves the essential spectral content of the operator. III. HOLOMORPHIC FAMILIES, KATO THEORY, AND ANALYTIC DRIFT In this section we analyse the analytic dependence of the drifted operator Ds on the deformation parameter s . This is essential for the rigorous control of spectral invariants, resolvents, and functional calculi used throughout the DSRN programme. The appropriate mathematical framework is provided by Kato’s theory of holomorphic families of operators. A. Kato Type-(A) Families Let D be a self–adjoint operator with compact resolvent and let φ be a bounded self–adjoint operator on H. Consider the drift family Ds=esφDe−sφ. Definition 5 (Holomorphic Family of Type (A)).A family of operators {T ( s ) }s∈C is a holomorphic family of type (A) if: •the domain Dom(T(s)) is independent of s, •for all uin the common domain, the map s7→ T(s)uis holomorphic. Theorem 6 (Analyticity of the Drift Flow).The family {Ds}s∈C defines a holomorphic family of type (A) in the sense of Kato. Proof. Since φ is bounded, the exponential esφ is an entire function of s with values in bounded operators. For all u∈Dom(D), Dsu=esφD(e−sφu), and the map s7→ Dsuis holomorphic. Moreover, the domain Dom(Ds)=esφDom(D)is unitarily equivalent to Dom(D)and can be identified with a fixed domain via this equivalence. 6 B. Analytic Resolvents and Functional Calculus Holomorphic dependence implies analytic control of resolvents. For z∈C\σ(D), (Ds−z)−1=esφ(D−z)−1e−sφ,(III.1) and the map s7→ (Ds−z)−1is holomorphic in operator norm. As a consequence, for any bounded holomorphic function f on a neighbourhood of σ ( D ), the functional calculus satisfies f(Ds)=esφf(D)e−sφ.(III.2) This identity underlies the invariance of spectral actions and zeta functions under drift deformation. C. Spectral Projections and Analytic Continuation Let λ be an isolated eigenvalue of D with finite multiplicity. The associated spectral projection Pλ(D) = 1 2πi IΓ (D−z)−1dz (III.3) extends analytically to the drifted family, Pλ(Ds)=esφPλ(D)e−sφ.(III.4) Eigenvalues are therefore constant along the drift flow, while eigenvectors vary analytically in s. D. Generator of the Drift Flow Differentiating Dswith respect to syields d dsDs= [φ, Ds].(III.5) This commutator equation shows that the drift flow is generated by an inner derivation on the algebra of operators. It provides the analytic backbone for interpreting drift as a spectral flow, renormalisation group evolution, or geometric deformation. Remark 7. The analyticity of the drift flow guarantees that no level crossing or spectral instability can occur under finite drift deformations. 7 E. Consequences for Spectral Geometry The Kato–holomorphic nature of the drift family has several immediate consequences: •spectral invariants depending holomorphically on Dare drift invariant; •heat kernels and zeta functions admit analytic continuation in s; •perturbative expansions in sare mathematically controlled; •functional traces remain well defined under drift. These results provide the analytic foundation required for the geometric, cosmological, holographic, and quantum applications developed in earlier reports. IV. PSEUDODIFFERENTIAL ORDER, BCH EXPANSION, AND DRIFT CORRECTIONS In this section we establish precise control over the pseudodifferential order of drift corrections and formalise the Baker–Campbell–Hausdorff (BCH) expansion underlying the drift deformation. These results ensure that drift geometry preserves the analytic class of Dirac–type operators and that all induced corrections are of strictly lower order. A. Pseudodifferential Operators and Order Let D be a Dirac–type operator of order one acting on sections of a vector bundle over a compact manifold. Then D belongs to the pseudodifferential class Ψ 1 , and its square D2 belongs to Ψ 2 . Bounded operators and multiplication by smooth functions belong to Ψ0. Lemma 8 (Order of Commutators).Let A∈ Ψ m and B∈ Ψ 0 . Then the commutator [ B, A ] ∈ Ψm−1. Proof. This is a standard result in pseudodifferential calculus, following from the symbol expansion and the Leibniz rule for symbols. B. BCH Expansion of the Drifted Operator Let φ∈Ψ0be a bounded self–adjoint operator. The drifted operator is given by Ds=esφDe−sφ.(IV.1) 8 Using the BCH formula, one obtains the formal expansion Ds=D+s[φ, D] + s2 2[φ, [φ, D]] + s3 6[φ, [φ, [φ, D]]] + · · · .(IV.2) Theorem 9 (Order Reduction in the BCH Tower).If D∈Ψ1and φ∈Ψ0, then: •[φ, D]∈Ψ0, •[φ, [φ, D]] ∈Ψ−1, •in general, the n–th iterated commutator belongs to Ψ1−n. Proof. By repeated application of the previous lemma, each commutator with a Ψ 0 operator lowers the order by one. C. Stability of the Dirac Class Since all higher–order BCH corrections are of order zero or lower, the principal symbol of Ds coincides with that of D . In particular, ellipticity and the Dirac–type character are preserved under drift. Corollary 10 (Ellipticity Preservation).If Dis elliptic, then Dsis elliptic for all s∈R. D. Drift Corrections to the Square The square of the drifted operator admits the expansion D2 s=D2+s{D, [φ, D]}+s2[φ, D]2+O(s3),(IV.3) where {·,·} denotes the anticommutator. The leading correction is of order one, while the quadratic term is of order zero. All higher–order terms are strictly lower order. E. Consequences for Heat Kernel Asymptotics Since the leading symbol of D2 s coincides with that of D2 , the heat kernel asymptotic expansion remains valid under drift deformation. Drift corrections modify only the lower Seeley–DeWitt coefficients, leaving the leading geometric terms unchanged. This justifies all heat–kernel computations used throughout the DSRN programme. Remark 11. The BCH expansion is not merely formal: it is asymptotic in the sense of pseudodifferential calculus and controlled term by term. 9 F. Summary We have shown that: •drift corrections lower pseudodifferential order systematically; •the Dirac class is preserved under drift; •ellipticity and principal symbols are invariant; •all geometric and spectral corrections are analytically controlled. These results complete the microlocal foundation of drift geometry. V. HEAT KERNEL, ZETA FUNCTIONS, AND SPECTRAL INVARIANTS UNDER DRIFT We analyse the behaviour of heat kernels, zeta functions, and associated spectral invariants under drift deformation. These objects play a central role in spectral geometry, index theory, and the spectral action. The results of this section establish their invariance and analytic control within drift geometry. A. Heat Kernel under Similarity Transformations Let D be a self–adjoint Dirac–type operator with compact resolvent, and let Ds = esφDe−sφ be its drifted counterpart. Consequently, for the pure heat trace defined by functional calculus and cyclicity of the trace, one has Tre−tD2 s= Tre−tD2, t > 0.(V.1) This invariance holds at the level of global spectral traces under bounded similarity transformations. It does not preclude the appearance of drift dependence in effective actions derived from lower–order operator data, local geometric decompositions, or restricted spectral sectors.