Krein-type Spectral Shift Formula

• Krein-type spectral shift formulas are identities that relate changes in the spectrum of operator pairs to perturbation determinants and scattering phases.
• They extend the classical Lifshits–Krein trace formula by incorporating resolvent-comparable, relatively trace-class, and non-self-adjoint settings using canonical normalizations.
• Applications span Schrödinger operators, boundary-value problems, obstacle scattering, and index theory, providing unified insights into spectral flow and perturbation analysis.

Searching arXiv for recent and foundational papers on Krein-type spectral shift formulas. The Krein-type spectral shift formula is the family of trace and determinant identities that encode how the spectrum of an operator pair changes under perturbation through a spectral shift function (SSF), usually denoted $\xi$. In the classical self-adjoint trace-class setting, $\xi(\lambda;H_1,H_0)$ is characterized by the Lifshits–Krein trace formula

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda,$$

together with a normalization convention, and is linked to perturbation determinants and scattering phases (Azamov et al., 2016, Gesztesy et al., 2010). Modern usage of “Krein-type” extends this framework far beyond the original trace-class case, including resolvent-comparable pairs, relatively trace-class perturbations, higher-order and relative Schatten settings, non-self-adjoint perturbations, contraction and dissipative pairs, Banach-space variants, and boundary-extension formulations via Krein resolvent identities (Azamov et al., 2016, Nuland et al., 2021, Bruneau et al., 23 Mar 2026, Malamud et al., 26 Mar 2026, Mirotin, 2018).

1. Classical formulation and normalization

For self-adjoint operators $H_0$ and $H_1$ on a Hilbert space, the classical spectral shift function is defined in the trace-class regime $H_1-H_0\in\mathfrak{S}_1$ and satisfies Krein’s trace identity

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda$$

for appropriate test functions $\varphi$ (Azamov et al., 2016, Gesztesy et al., 2010, Carey et al., 2015). In resolvent form this becomes

$$-\operatorname{Tr}\big((H_1-zI)^{-1}-(H_0-zI)^{-1}\big) = \int_{\mathbb{R}}\xi(\lambda;H_1,H_0)(\lambda-z)^{-2}\,d\lambda, \qquad z\in\mathbb{C}\setminus\mathbb{R},$$

which is one of the canonical entry points for generalization (Gesztesy et al., 2010, Connard et al., 2022).

In general, the trace identity determines $\xi$ only up to an additive constant. Several normalization mechanisms appear in the literature summarized here. In the resolvent-comparable self-adjoint framework, Birman–Solomyak spectral averaging provides a canonical normalization: $\xi(\lambda;H_1,H_0)$0 with $\xi(\lambda;H_1,H_0)$1 (Azamov et al., 2016). In relatively trace-class problems for asymptotic pairs $\xi(\lambda;H_1,H_0)$2, an invariance principle is used: $\xi(\lambda;H_1,H_0)$3 and the residual additive ambiguity is fixed by gap conditions near $\xi(\lambda;H_1,H_0)$4 (Gesztesy et al., 2010). For lower semibounded self-adjoint pairs, a standard normalization is $\xi(\lambda;H_1,H_0)$5 to the left of the joint spectral bottom (Carey et al., 2015, Allan et al., 2019).

A perturbation-determinant representation is another central classical feature. In the trace-class self-adjoint case, if

$\xi(\lambda;H_1,H_0)$6

then

$\xi(\lambda;H_1,H_0)$7

equivalently through boundary values of $\xi(\lambda;H_1,H_0)$8 (Bruneau et al., 23 Mar 2026, Carey et al., 2015). This determinant perspective persists, with modifications, in many generalized settings.

2. Resolvent-comparable extensions and the canonical ac/singular split

A major extension replaces the trace-class perturbation hypothesis by resolvent comparability: $\xi(\lambda;H_1,H_0)$9 for one, hence every, nonreal $$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda,$$0 (Azamov et al., 2016). In this framework the Birman–Solomyak definition yields a spectral shift measure that is absolutely continuous, and the trace formula follows by integrating a chain rule under the trace (Azamov et al., 2016).

The decisive refinement developed for resolvent-comparable pairs is the canonical decomposition

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda,$$1

where the absolutely continuous and singular parts are defined by replacing the full spectral measure by its absolutely continuous or singular part inside the infinitesimal spectral shift measure (Azamov et al., 2016). Explicitly,

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda,$$2

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda,$$3

For almost every $$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda,$$4, the absolutely continuous density admits the stationary formula

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda,$$5

(Azamov et al., 2016).

The singular part is especially distinctive. Under the resolvent-comparable hypotheses used in "Singular spectral shift function for Schrödinger operators" (Azamov et al., 2016), the singular spectral shift measure is absolutely continuous and its density is integer-valued almost everywhere: $$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda,$$6 This is obtained by combining the full Birman–Krein determinant formula

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda,$$7

with its absolutely continuous variant

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda,$$8

which forces $$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda,$$9 (Azamov et al., 2016). This exhibits a Krein-type formula in which scattering determines not only the total shift but also isolates an integer-valued singular component.

The same paper proves two nontrivial identifications of the singular part. First,

$H_0$0

so the singular SSF equals the total resonance index (Azamov et al., 2016). Second,

$H_0$1

where $H_0$2 is the singular part of Pushnitski’s $H_0$3-invariant, defined through scattering eigenphase flow and independent of the angle variable (Azamov et al., 2016). In the earlier trace-class setting, the angle-independence of $H_0$4 and the equality

$H_0$5

were established in "Singular spectral shift and Pushnitski $H_0$6-invariant" (Azamov, 2010).

For Schrödinger operators,

$H_0$7

with $H_0$8 bounded measurable real-valued and $H_0$9 bounded integrable real-valued on $H_1$0, the resolvent-comparable hypothesis holds for $H_1$1 but not in general for $H_1$2 under these assumptions (Azamov et al., 2016). This dimensional restriction is a structural limitation of that realization of the Krein-type theory.

3. Relatively trace-class perturbations, index theory, and spectral flow

A different generalization concerns perturbations that are not themselves trace class but are relatively trace class. In the framework of "The index formula and the spectral shift function for relatively trace class perturbations" (Gesztesy et al., 2010), one studies a path

$H_1$3

with $H_1$4 self-adjoint, $H_1$5 a.e., and

$H_1$6

(Gesztesy et al., 2010). These assumptions imply the existence of asymptotes $H_1$7 with

$H_1$8

(Gesztesy et al., 2010).

The associated first-order operator

$H_1$9

leads to second-order operators

$H_1-H_0\in\mathfrak{S}_1$0

and the central resolvent trace identity becomes

$H_1-H_0\in\mathfrak{S}_1$1

with

$H_1-H_0\in\mathfrak{S}_1$2

(Gesztesy et al., 2010). This is the core Krein-type relation from which the paper derives a Pushnitski-type transform linking the SSF for $H_1-H_0\in\mathfrak{S}_1$3 to the SSF for $H_1-H_0\in\mathfrak{S}_1$4: $H_1-H_0\in\mathfrak{S}_1$5 (Gesztesy et al., 2010).

This framework has direct index-theoretic consequences. If $H_1-H_0\in\mathfrak{S}_1$6, then $H_1-H_0\in\mathfrak{S}_1$7 is Fredholm and

$H_1-H_0\in\mathfrak{S}_1$8

(Gesztesy et al., 2010). Moreover,

$H_1-H_0\in\mathfrak{S}_1$9

and also

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda$$0

(Gesztesy et al., 2010).

The survey "The Spectral shift function and the Witten index" (Carey et al., 2015) places these formulas into a broader Krein-type context. It reiterates that when $$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda$$1,

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda$$2

and shows that in the non-Fredholm case, provided $$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda$$3 is a right and left Lebesgue point of $$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda$$4, the resolvent-regularized Witten index is

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda$$5

(Carey et al., 2015). A plausible implication is that the Krein-type spectral shift formula functions այստեղ not only as a perturbative trace identity but as a unifying invariant across Fredholm index theory, spectral flow, and regularized index notions.

4. Determinant formulas, boundary conditions, and Krein resolvent identities

A second major line of development uses Krein resolvent formulas for self-adjoint extensions. In one-dimensional Schrödinger theory with coupled boundary conditions, "Weak convergence of spectral shift functions revisited" (Connard et al., 2022) derives an explicit rank-two Krein-type resolvent identity. For the finite-interval operators $$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda$$6 and Dirichlet reference operators $$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda$$7,

$$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda$$8

where $$\operatorname{Tr}\big(\varphi(H_1)-\varphi(H_0)\big)=\int_{\mathbb{R}}\varphi'(\lambda)\,\xi(\lambda;H_1,H_0)\,d\lambda$$9 is rank two and is written using solutions $\varphi$0 and a $\varphi$1 matrix $\varphi$2 (Connard et al., 2022). The paper interprets this in boundary-triplet form as

$\varphi$3

with $\varphi$4 (Connard et al., 2022). In this setting, the Birman–Krein scattering formula is not used; the analysis proceeds through resolvent identities and trace-ideal control alone (Connard et al., 2022).

For singular Sturm–Liouville operators, "Explicit Krein Resolvent Identities for Singular Sturm-Liouville Operators with Applications to Bessel Operators" (Allan et al., 2019) gives rank-one and rank-two formulas in terms of boundary condition bases and the Lagrange bracket. In the one-limit-circle case,

$\varphi$5

with

$\varphi$6

so the resolvent difference is rank one (Allan et al., 2019). Consequently,

$\varphi$7

(Allan et al., 2019). In the Bessel application, this leads to an explicit perturbation determinant $\varphi$8 and

$\varphi$9

followed by the spectral shift representation

$$-\operatorname{Tr}\big((H_1-zI)^{-1}-(H_0-zI)^{-1}\big) = \int_{\mathbb{R}}\xi(\lambda;H_1,H_0)(\lambda-z)^{-2}\,d\lambda, \qquad z\in\mathbb{C}\setminus\mathbb{R},$$0

(Allan et al., 2019).

A more geometric realization appears for Laplacians on Euclidean surfaces with conical singularities. In "Krein formula and S-matrix for Euclidean Surfaces with Conical Singularities" (Hillairet et al., 2010), the Weyl function is a boundary $$-\operatorname{Tr}\big((H_1-zI)^{-1}-(H_0-zI)^{-1}\big) = \int_{\mathbb{R}}\xi(\lambda;H_1,H_0)(\lambda-z)^{-2}\,d\lambda, \qquad z\in\mathbb{C}\setminus\mathbb{R},$$1-matrix $$-\operatorname{Tr}\big((H_1-zI)^{-1}-(H_0-zI)^{-1}\big) = \int_{\mathbb{R}}\xi(\lambda;H_1,H_0)(\lambda-z)^{-2}\,d\lambda, \qquad z\in\mathbb{C}\setminus\mathbb{R},$$2 rather than a scattering matrix in the usual asymptotic sense. For a regular self-adjoint extension $$-\operatorname{Tr}\big((H_1-zI)^{-1}-(H_0-zI)^{-1}\big) = \int_{\mathbb{R}}\xi(\lambda;H_1,H_0)(\lambda-z)^{-2}\,d\lambda, \qquad z\in\mathbb{C}\setminus\mathbb{R},$$3 relative to the Friedrichs extension $$-\operatorname{Tr}\big((H_1-zI)^{-1}-(H_0-zI)^{-1}\big) = \int_{\mathbb{R}}\xi(\lambda;H_1,H_0)(\lambda-z)^{-2}\,d\lambda, \qquad z\in\mathbb{C}\setminus\mathbb{R},$$4, the paper obtains

$$-\operatorname{Tr}\big((H_1-zI)^{-1}-(H_0-zI)^{-1}\big) = \int_{\mathbb{R}}\xi(\lambda;H_1,H_0)(\lambda-z)^{-2}\,d\lambda, \qquad z\in\mathbb{C}\setminus\mathbb{R},$$5

and writes

$$-\operatorname{Tr}\big((H_1-zI)^{-1}-(H_0-zI)^{-1}\big) = \int_{\mathbb{R}}\xi(\lambda;H_1,H_0)(\lambda-z)^{-2}\,d\lambda, \qquad z\in\mathbb{C}\setminus\mathbb{R},$$6

(Hillairet et al., 2010). The associated step-function spectral shift has jumps equal to eigenvalue multiplicity differences (Hillairet et al., 2010). The same determinant controls zeta-regularized determinant comparisons: $$-\operatorname{Tr}\big((H_1-zI)^{-1}-(H_0-zI)^{-1}\big) = \int_{\mathbb{R}}\xi(\lambda;H_1,H_0)(\lambda-z)^{-2}\,d\lambda, \qquad z\in\mathbb{C}\setminus\mathbb{R},$$7 (Hillairet et al., 2010). This suggests that in extension theory the Krein-type spectral shift formula is naturally intertwined with Weyl functions, boundary data, and zeta determinants rather than only with scattering on an absolutely continuous spectrum.

5. Specialized realizations in scattering and low-dimensional models

In scattering theory, the Krein-type spectral shift formula often appears as a combined trace/scattering relation. For planar obstacle scattering, "The spectral shift function for planar obstacle scattering at low energy" (McGillivray, 2011) defines the SSF by an invariance principle from semigroups and proves

$$-\operatorname{Tr}\big((H_1-zI)^{-1}-(H_0-zI)^{-1}\big) = \int_{\mathbb{R}}\xi(\lambda;H_1,H_0)(\lambda-z)^{-2}\,d\lambda, \qquad z\in\mathbb{C}\setminus\mathbb{R},$$8

for a specific class of test functions (McGillivray, 2011). The Birman–Krein formula takes the form

$$-\operatorname{Tr}\big((H_1-zI)^{-1}-(H_0-zI)^{-1}\big) = \int_{\mathbb{R}}\xi(\lambda;H_1,H_0)(\lambda-z)^{-2}\,d\lambda, \qquad z\in\mathbb{C}\setminus\mathbb{R},$$9

(McGillivray, 2011). In two dimensions the threshold behavior is logarithmic: $\xi$0 with $\xi$1 (McGillivray, 2011). This low-energy expansion is then transferred to large-time asymptotics of the pinned Wiener sausage (McGillivray, 2011).

For obstacle assemblies, "A relative trace formula for obstacle scattering" (Hanisch et al., 2020) introduces a relative spectral shift

$\xi$2

so interior eigenvalue terms cancel (Hanisch et al., 2020). The corresponding relative trace is encoded by a holomorphic function $\xi$3 satisfying

$\xi$4

on the positive real axis, and

$\xi$5

for a large sectorial class of test functions (Hanisch et al., 2020). The interaction part of the shift is represented by a boundary-layer determinant

$\xi$6

(Hanisch et al., 2020). This is a relative Birman–Krein mechanism adapted to multi-obstacle interaction.

In periodic or finite-gap settings, the terminology “Krein-type” can become more formal. "Spectral Shift Functions of Lamé Operators" (Idiong et al., 25 Mar 2025) reviews the classical Lifshits–Krein formula, perturbation determinants, and Birman–Krein relation, then computes explicit Green-function-based phase formulas for Lamé and Brioschi–Halphen operators through distributional Fourier methods (Idiong et al., 25 Mar 2025). The paper emphasizes that for periodic backgrounds one typically works with per-period traces or relative normalizations because $\xi$7 is usually not trace class globally (Idiong et al., 25 Mar 2025).

For first-order ODEs with $\xi$8-periodic boundary conditions, "Hill-type formula and Krein-type trace formula for $\xi$9-periodic solutions in ODEs" (Hu et al., 2015) uses conditional Fredholm determinants rather than an SSF on the real line. The central identity compares

$\xi(\lambda;H_1,H_0)$00

to the monodromy determinant $\xi(\lambda;H_1,H_0)$01 (Hu et al., 2015). Differentiation yields trace formulas for powers of

$\xi(\lambda;H_1,H_0)$02

that the paper explicitly describes as a non-self-adjoint analogue of the Hamiltonian case (Hu et al., 2015). This suggests that “Krein-type” can denote the structural role of logarithmic-derivative trace identities even when no scalar SSF is introduced.

6. Generalizations beyond the classical self-adjoint Hilbert-space setting

Several recent directions modify either the operator class or the perturbation class while retaining the Krein paradigm.

For higher-order formulas under relative Schatten assumptions, "Spectral shift for relative Schatten class perturbations" (Nuland et al., 2021) considers self-adjoint $\xi(\lambda;H_1,H_0)$03 and bounded self-adjoint $\xi(\lambda;H_1,H_0)$04 with

$\xi(\lambda;H_1,H_0)$05

It proves the existence of a real-valued higher-order spectral shift function $\xi(\lambda;H_1,H_0)$06 such that for $\xi(\lambda;H_1,H_0)$07 in a concrete class $\xi(\lambda;H_1,H_0)$08,

$\xi(\lambda;H_1,H_0)$09

(Nuland et al., 2021). The function is unique up to a polynomial of degree at most $\xi(\lambda;H_1,H_0)$10 and satisfies weighted integrability bounds (Nuland et al., 2021). This is the higher-order analogue of the Krein–Koplienko–Peller–Skripka trace formula in the relative setting.

For Banach spaces, "Lifshitz-Krein trace formula for Hirsch functiuonal calculus on Banach spaces" (Mirotin, 2018) defines a spectral shift function for pairs of nonpositive or negative operators under nuclear perturbations. If $\xi(\lambda;H_1,H_0)$11 and $\xi(\lambda;H_1,H_0)$12, then for negative complete Bernstein functions $\xi(\lambda;H_1,H_0)$13 satisfying the paper’s integrability assumptions,

$\xi(\lambda;H_1,H_0)$14

where

$\xi(\lambda;H_1,H_0)$15

for $\xi(\lambda;H_1,H_0)$16 and is analytically continued to a sector (Mirotin, 2018). This is a genuine Banach-space Krein-type formula, but it is not based on self-adjoint spectral measures.

For non-self-adjoint perturbations of self-adjoint operators, "The Spectral Shift Function for Non-Self-Adjoint Perturbations" (Bruneau et al., 23 Mar 2026) defines the SSF as a distribution via

$\xi(\lambda;H_1,H_0)$17

for $\xi(\lambda;H_1,H_0)$18 under hypotheses controlling non-real eigenvalues, resolvent growth, and relatively trace-class behavior (Bruneau et al., 23 Mar 2026). The key jump formula is

$\xi(\lambda;H_1,H_0)$19

in the sense of distributions (Bruneau et al., 23 Mar 2026). The determinant representation becomes

$\xi(\lambda;H_1,H_0)$20

with $\xi(\lambda;H_1,H_0)$21 (Bruneau et al., 23 Mar 2026). Here $\xi(\lambda;H_1,H_0)$22 may be complex-valued, spectral singularities produce principal-value and delta-derivative terms in $\xi(\lambda;H_1,H_0)$23, and the classical unitary Birman–Krein identity is no longer expected in unchanged form (Bruneau et al., 23 Mar 2026).

A parallel non-self-adjoint direction concerns contractions and dissipative operators. "Real-valued spectral shift functions for contractions and dissipative operators" (Malamud et al., 2024) and its expanded version (Malamud et al., 26 Mar 2026) show that for contractions $\xi(\lambda;H_1,H_0)$24 with $\xi(\lambda;H_1,H_0)$25, there exists an integrable SSF $\xi(\lambda;H_1,H_0)$26 on $\xi(\lambda;H_1,H_0)$27 such that

$\xi(\lambda;H_1,H_0)$28

for $\xi(\lambda;H_1,H_0)$29 in the operator-Lipschitz disk-algebra class (Malamud et al., 2024, Malamud et al., 26 Mar 2026). Unlike the self-adjoint case, $\xi(\lambda;H_1,H_0)$30 is nonunique: adding any $\xi(\lambda;H_1,H_0)$31 function yields another SSF (Malamud et al., 2024, Malamud et al., 26 Mar 2026). Real-valued representatives require extra structure. For maximal dissipative operators one has the rational-function trace formula

$\xi(\lambda;H_1,H_0)$32

under trace-class or resolvent-difference hypotheses, but a real-valued integrable SSF can fail to exist unless additional conditions hold; in particular, if $\xi(\lambda;H_1,H_0)$33 and $\xi(\lambda;H_1,H_0)$34, then no real-valued integrable SSF exists (Malamud et al., 26 Mar 2026).

Taken together, these developments show that the phrase “Krein-type spectral shift formula” now denotes a broad perturbative architecture rather than a single theorem. The stable core consists of three interlocking elements: a trace formula, a logarithmic-derivative or perturbation-determinant identity, and a notion of spectral phase or counting invariant. What changes across settings is the regularity class of the perturbation, the normalization of $\xi(\lambda;H_1,H_0)$35, the ambient functional calculus, and the status of real-valuedness, absolute continuity, and scattering interpretation (Azamov et al., 2016, Gesztesy et al., 2010, Nuland et al., 2021, Bruneau et al., 23 Mar 2026, Malamud et al., 26 Mar 2026).