DRSN X: Mathematical Foundations of Drift Geometry (De Rerum Spectrale Natura, Report X, Version 2.0)

DRSN X: MATHEMATICAL FOUNDATIONS OF DRIFT GEOMETRY De Rerum Spectrale Natura series REPORT X (Version 2.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 •Drift geometry formulated as a bounded similarity flow on unbounded Dirac operators. •Complete control of domains, self–adjointness, and spectral invariants under drift. •Rigorous Kato–holomorphic structure of the drift family. •Microlocal and pseudodifferential analysis of BCH drift corrections. •Heat kernel, zeta functions, and spectral action shown to be invariant under drift. •Mathematical closure of the DRSN programme and separation of rigid versus effective dynamics. DRSN 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 DRSN 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. ∗jp[email protected] 3 CONTENTS I. Introduction 5 II. Functional–Analytic Framework of Drift Geometry 6 A. Hilbert Space Setting 6 B. Drift as a Similarity Flow 6 C. Basic Structural Properties 6 III. Holomorphic Families, Kato Theory, and Analytic Drift 7 A. Kato Type-(A) Families 7 B. Analytic Resolvents and Functional Calculus 8 C. Spectral Projections and Analytic Continuation 8 D. Generator of the Drift Flow 8 E. Consequences for Spectral Geometry 9 IV. Pseudodifferential Order, BCH Expansion, and Drift Corrections 9 A. Pseudodifferential Operators and Order 9 B. BCH Expansion of the Drifted Operator 9 C. Stability of the Dirac Class 10 D. Drift Corrections to the Square 10 E. Consequences for Heat Kernel Asymptotics 10 F. Summary 11 V. Heat Kernel, Zeta Functions, and Spectral Invariants under Drift 11 A. Heat Kernel under Similarity Transformations 11 B. Heat Kernel Asymptotics 12 C. Spectral Zeta Functions 12 D. Eta Invariants and Spectral Asymmetry 12 E. Spectral Action Invariance 13 F. Consequences for Geometry and Physics 13 VI. Closure of Drift Geometry, Open Problems, and Outlook 14 A. Mathematical Closure of Drift Geometry 14 B. Conceptual Significance 14 4 C. Open Mathematical Problems 15 D. Relation to the DRSN Programme 15 E. Outlook 16 Appendices 17 A. Summary of Core Results 17 References 18 5 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 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 DRSN programme and as an independent contribution to operator theory and spectral geometry. This report formalises the operator–theoretic layer used throughout the DRSN series, including the drift potential and cosmological sector (DRSN I), unified/worldsheet formulations (DRSN II), supersymmetric and higher-dimensional extensions (DRSN III–V), brane and holographic constructions (DRSN VI–VII), quantum drift geometry (DRSN VIII), and particle–string unification (DRSN IX) [1–9]. About this report. This work constitutes Report X of the DRSN series (De Rerum Spectrale Natura), a sequence of independent but thematically unified studies on drifted spectral geometry. The present report provides the mathematical and functional–analytic backbone of the DSRN programme, upon which all geometric, physical, and quantum applications developed in Reports I–IX rely. In contrast to previous reports, all physical interpretations are deliberately set aside. The focus here is exclusively on unbounded operators, similarity flows, spectral invariants, and analytic control. This report fixes the methodological boundary between rigid spectral data and effective dynamics used throughout the DSRN series. The DRSN series is developed within an open research community on spectral geometry and fundamental physics. Related materials, preprints, and versioned updates are archived at https: //zenodo.org/communities/dsrn/. 6 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. Background on closed operators, self–adjointness and spectral theory is standard [ 10 ]. This setting includes Dirac operators of spectral triples in noncommutative geometry [11]. 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. Theorem 4 (Spectral Invariance).For all s∈R, σ(Ds)=σ(D), and the resolvents satisfy (Ds−z)−1=esφ(D−z)−1e−sφ. 7 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 DRSN programme. The appropriate mathematical framework is provided by Kato’s theory of holomorphic families of operators. We follow the standard operator–theoretic framework of holomorphic families in the sense of Kato [12]. 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. 8 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 ) −1 is holomorphic in operator norm. Semigroup and resolvent control in this setting is standard in the theory of linear evolution equations [13]. 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. 9 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) 16 E. Outlook Drift geometry opens a mathematically controlled path for deforming geometric structures without compromising spectral invariants. Beyond its applications in physics, it suggests new directions in operator theory, spectral geometry, and noncommutative analysis. In particular, drift flows may provide a new organising principle for families of operators, renormalisation–like structures, and analytic deformations in global analysis. We conclude that drift geometry constitutes a well–defined and robust mathematical framework, whose functional–analytic foundations are now fully established. In particular, the emergence of drift–dependent effective potentials in earlier reports is fully compatible with the present analysis, as it relies on effective geometric functionals rather than on invariant spectral traces. 17 APPENDICES Appendix A: Summary of Core Results For ease of reference, we summarise the main mathematical statements proved in this report: •drift flows preserve self–adjointness and domain stability; •the drift family is holomorphic of type (A); •spectra, resolvents, and spectral projections are invariant; •BCH expansions lower pseudodifferential order systematically; •heat kernel coefficients, zeta functions, and determinants are preserved. 18 REFERENCES [1] J. Pinho-da-Cruz, DRSN I: A Unified Geometric Framework for Matter, Torsion and Dark Energy, August 2025. Version 3.0, doi: 10.5281/zenodo.17981231. [2] J. Pinho-da-Cruz, DRSN II: Drift Geometry and the Unified Spectral Operator, September 2025. Version 2.0, doi: 10.5281/zenodo.17982131. [3] J. Pinho-da-Cruz, DRSN III: Supersymmetric Drift Geometry and the Spectral Master Operator, September 2025. Version 2.0, doi: 10.5281/zenodo.17986205. [4] J. Pinho-da-Cruz, DRSN IV: Drifted M-Theory Spectral Geometry, September 2025. Version 2.0, doi: 10.5281/zenodo.17987598. [5] J. Pinho-da-Cruz, DRSN V: Drifted Compactifications and Spectral Moduli Dynamics, September 2025. Version 2.0, doi: 10.5281/zenodo.17990670. [6] J. Pinho-da-Cruz, DRSN VI: Drifted Brane Geometry and Spectral Worldvolume Dynamics, September 2025. Version 2.0, doi: 10.5281/zenodo.17992350. [7] J. Pinho-da-Cruz, DRSN VII: Drifted Holographic Geometry and Spectral Gauge–Gravity Duality, September 2025. Version 2.0, doi: 10.5281/zenodo.17992722. [8] J. Pinho-da-Cruz, DRSN VIII: Quantum Drift Geometry, September 2025. Version 2.0, doi: 10.5281/zenodo.17997240. [9] J. Pinho-da-Cruz, DRSN IX: Spectral Standard Model and Drifted Strings, September 2025. Version 2.0, doi: 10.5281/zenodo.17997240. [10] M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, New York (1975). [11] A. Connes, Noncommutative Geometry, Academic Press, San Diego (1994). [12] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1995). [13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983). [14] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge (1989). [15] P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, CRC Press, Boca Raton (1995).