Symplectic Resolvent Difference Formula

• The symplectic resolvent difference formula is a precise mathematical expression that relates two operator resolvents through boundary symplectic data.
• It underpins key results in spectral theory and operator algebras, notably in Sturm–Liouville systems, discrete models, and canonical block operator frameworks.
• Its applications extend to computing spectral shift functions, asymptotic eigenvalue behavior, and facilitating topological recursion in mathematical physics.

A symplectic resolvent difference formula expresses the difference between the resolvents of two operators or two realizations of a system—such as self-adjoint extensions, boundary conditions, or perturbations—in terms of boundary symplectic data. This structure is fundamental in spectral theory, mathematical physics, and the analysis of canonical systems, with deep connections to the theory of operator algebras, Sturm–Liouville (continuous and discrete) systems, and topological recursion. The symplectic structure governs both the algebraic and analytic properties of these difference formulas, making them central constructs in modern operator theory.

1. Universal Form: Symplectic Difference Identity in Operator Algebras

For canonical commutation relations, the resolvent algebra $\mathcal{R}(X,\sigma)$ generated by operators $R(\lambda, f) = (i\lambda - \phi(f))^{-1}$ (with $\phi(f)$ self-adjoint, $[\phi(f), \phi(g)] = i\,\sigma(f,g)\,\mathbf{1}$ and real symplectic form $\sigma$) encapsulates the symplectic resolvent-difference at the algebraic level. The foundational resolvent-difference identity is

$$R(\lambda, f) - R(\mu, f) = i\,(\mu - \lambda)\,R(\lambda, f)\,R(\mu, f).$$

This expresses the difference between resolvents at different spectral parameters for the same field, directly encoding the symplectic structure (Bauer et al., 2022).

Additionally, the commutator identity

$$[R(\lambda, f), R(\mu, g)] = i\,\sigma(f, g)\,R(\lambda, f)\,R(\mu, g)^2\,R(\lambda, f)$$

illustrates how the symplectic form $\sigma$ governs the non-commutativity of the resolvent generators.

2. Kreĭn-Type and Boundary-Data Symplectic Resolvent Formulas

In the context of self-adjoint extensions of symmetric differential operators (classical continuous and discrete Sturm–Liouville problems), the difference between resolvents of two extensions is explicitly encoded by symplectic pairings of boundary data. For continuous regular/irregular Sturm–Liouville operators,

$$(H_\Theta - z)^{-1} - (H_0 - z)^{-1} = -\sum_{j,k=1}^2 \frac{(u_k(z),\,\cdot\,)\,u_j(\bar z)}{\omega(Y u_j(\bar z), Y u_k(z))}$$

where $u_j(z)$ span the deficiency space at $z$, $Y$ denotes the trace map (for boundary data), and $\omega$ is the standard boundary symplectic form $$\omega((\alpha, \beta), (\alpha', \beta')) = \alpha\beta' - \alpha'\beta$$ (Clark et al., 2012, Allan et al., 2019).

For the discrete symplectic case, the Kreĭn resolvent difference formula generalizes as

$$(T_\alpha - \lambda I)^{-1} - (T_{\alpha_0} - \lambda I)^{-1} = \gamma(\lambda) \bigl[U(\alpha) - M_+(\lambda)\bigr]^{-1} \gamma(\bar{\lambda})^*$$

with $U(\alpha) = \alpha J \alpha^*$ (boundary parameter), $M_+(\lambda)$ the Weyl–Titchmarsh function, and $\gamma(\lambda)$ the $\ell^2$-valued Weyl solution (Zemánek, 2024).

3. Symplectic Resolvent Difference in Canonical Symplectic and Block Operator Systems

For abstract symplectic operator systems, as in canonical block operator formalism, the symplectic resolvent difference formula takes the form

$$R_t(\lambda) - R_s(\lambda) = \tau \bigl(T_s R_t(\bar{\lambda})\bigr)^* J T_t R_s(\lambda)$$

where $\tau$ is the swap involution, $J$ the canonical symplectic matrix, $T_t$ the trace operator to a Lagrangian subspace, and $(\cdot)^*$ the Hilbert space adjoint. This formula quantifies the resolvent change under boundary or parameter variation, and encodes all contributions (potential perturbations, change of Cauchy data plane, boundary adiabatic motion) through explicit symplectic data (Curran et al., 2 Jan 2026).

The symplectic difference also underpins first-order asymptotic expansions for eigenvalue curves (the symplectic Rayleigh–Hadamard formula), with the Maslov crossing form appearing as the symplectic analogue of a perturbation matrix.

4. Symplectic Resolvent Difference in Topological Recursion

In the context of topological recursion for spectral curves $(x, y)$, the symplectic structure governs the transformation of correlators under $x \leftrightarrow y$ swaps. The symplectic resolvent difference at the one-point function level reads

$$\Delta W(z) = W_{0,1}(z) - W^\vee_{0,1}(z) = y\,dx - x\,dy = d(xy)$$

capturing an exact form anomaly due to the lack of manifest $x \leftrightarrow y$-invariance at the level of correlators, while symplectic invariants $\mathcal{F}^g$ retain full invariance. For higher correlators, a combinatorial graph sum (the symplectic swap) provides the general transformation (Hock, 2022).

5. Symplectic Difference and Spectral Data: Determinants, Traces, and Shift Functions

Symplectic resolvent difference formulas yield several analytic and spectral identities. In continuous Sturm–Liouville theory,

$$\operatorname{tr}\bigl((H_\Theta - z)^{-1} - (H_0 - z)^{-1}\bigr) = -\frac{d}{dz}\ln\det_{\mathbb{C}^2}\Lambda(z)$$

where $\Lambda(z)$ is the boundary data map. Furthermore, the spectral shift function is given by

$$\xi(\lambda) = \pi^{-1} \lim_{\varepsilon \downarrow 0} \Im \ln\det_{\mathbb{C}^2}\Lambda(\lambda + i\varepsilon).$$

These formulas relate boundary symplectic data to the spectral theory of pairs of self-adjoint extensions (Clark et al., 2012). For discrete systems the analogue is present through the dependence of the resolvent and spectrum on the Weyl–Titchmarsh and boundary parameters (Zemánek, 2024).

6. Summary Table of Key Symplectic Resolvent Difference Formulas

Context	Formula Structure	Main Symplectic Data
Abstract $C^*$-algebra	$$R(\lambda, f) - R(\mu, f) = i\,(\mu-\lambda)\,R(\lambda, f)\,R(\mu, f)$$	Symplectic form $\sigma$
Sturm–Liouville/ODE	$$\sum_{j,k}\frac{(u_k(z),\cdot)\,u_j(\bar z)}{\omega(Yu_j(\bar z), Y u_k(z))}$$	Boundary symplectic form $\omega$
Discrete symplectic	$$\gamma(\lambda)[U(\alpha) - M_+(\lambda)]^{-1}\gamma(\bar{\lambda})^*$$	Weyl–Titchmarsh $M_+$, $U(\alpha)$
Block canonical systems	$\tau (T_s R_t(\bar{\lambda}))^* J T_t R_s(\lambda)$	Swap, $J$, trace operators
Topological recursion	$y\,dx - x\,dy = d(xy)$	Symplectic swap in $x$-$y$ variables

7. Significance, Conceptual Role, and Applications

The symplectic resolvent difference formula encodes structural information about how operator spectra, boundary conditions, and algebraic relations transform under symplectic modifications. It is central in the analysis of canonical quantum systems, spectral and scattering theory (continuous and discrete systems), index and spectral flow computations, and in modern enumerative geometry via topological recursion. In all cases, the symplectic structure is not merely an auxiliary object but is essential to the analytic and algebraic properties of resolvent differences, manifesting in precise, closed-form identities that underlie trace formulas, perturbative expansions, and spectral invariants (Clark et al., 2012, Bauer et al., 2022, Allan et al., 2019, Zemánek, 2024, Curran et al., 2 Jan 2026, Hock, 2022).