---
title: Krein-type Spectral Shift Formula
url: https://www.emergentmind.com/topics/krein-type-spectral-shift-formula
type: topic
---

# Krein-type Spectral Shift Formula

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 [1608.04184], [1004.1582]. 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 [1608.04184], [2102.00090], [2603.21773], [2603.25242], [1805.01337].

## 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 $\varphi$ [1608.04184], [1004.1582], [1505.04895]. 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 [1004.1582], [2211.14970].

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:
\[
\xi(\varphi)=\int_0^1 \operatorname{Tr}\big(E_{H_r}(\operatorname{supp}\varphi)\,V\,\varphi(H_r)\big)\,dr,
\qquad H_r=H_0+rV,
\]
with $V=H_1-H_0$ [1608.04184]. In relatively trace-class problems for asymptotic pairs $(A_+,A_-)$, an invariance principle is used:
\[
\xi(\nu;A_+,A_-):=\xi(g(\nu);g(A_+),g(A_-)),\qquad g(x)=x(x+1)^{-1/2},
\]
and the residual additive ambiguity is fixed by gap conditions near $0$ [1004.1582]. For lower semibounded self-adjoint pairs, a standard normalization is $\xi(\lambda)=0$ to the left of the joint spectral bottom [1505.04895], [1908.05392].

A perturbation-determinant representation is another central classical feature. In the trace-class self-adjoint case, if
\[
D(z)=\det\big(I+V(H_0-z)^{-1}\big),
\]
then
\[
\xi(\lambda)=\frac{1}{\pi}\lim_{\varepsilon\downarrow 0}\arg D(\lambda+i\varepsilon),
\]
equivalently through boundary values of $\log D$ [2603.21773], [1505.04895]. 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:
\[
(H_1-z)^{-1}-(H_0-z)^{-1}\in\mathfrak{S}_1
\]
for one, hence every, nonreal $z$ [1608.04184]. 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 [1608.04184].

The decisive refinement developed for resolvent-comparable pairs is the canonical decomposition
\[
\xi=\xi^{(a)}+\xi^{(s)},
\]
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 [1608.04184]. Explicitly,
\[
\xi^{(a)}(\varphi)=\int_0^1 \operatorname{Tr}\big(E^{(a)}_{H_r}(\operatorname{supp}\varphi)\,V\,\varphi(H_r)\big)\,dr,
\]
\[
\xi^{(s)}(\varphi)=\int_0^1 \operatorname{Tr}\big(E^{(s)}_{H_r}(\operatorname{supp}\varphi)\,V\,\varphi(H_r)\big)\,dr.
\]
For almost every $\lambda$, the absolutely continuous density admits the stationary formula
\[
\xi^{(a)}(\lambda;\{H_r\})
=
\frac{1}{\pi}\lim_{y\to0+}\int_0^1
\operatorname{Tr}\big(R_{\lambda-iy}(H_r)\,V\,R_{\lambda+iy}(H_r)\big)\,dr
\]
[1608.04184].

The singular part is especially distinctive. Under the resolvent-comparable hypotheses used in "Singular spectral shift function for Schrödinger operators" [1608.04184], the singular spectral shift measure is absolutely continuous and its density is integer-valued almost everywhere:
\[
\xi^{(s)}(\lambda)\in\mathbb{Z}\quad\text{for a.e. }\lambda.
\]
This is obtained by combining the full Birman–Krein determinant formula
\[
\det S(\lambda;H_1,H_0)=\exp\big(-2\pi i\,\xi(\lambda;H_1,H_0)\big)
\]
with its absolutely continuous variant
\[
\det S(\lambda;H_1,H_0)=\exp\big(-2\pi i\,\xi^{(a)}(\lambda;\{H_r\})\big),
\]
which forces $\exp(-2\pi i\xi^{(s)}(\lambda))=1$ [1608.04184]. 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,
\[
\xi^{(s)}(\lambda;H_1,H_0)=\sum_{r_\lambda\in[0,1]}\operatorname{ind}_{\mathrm{res}}(\lambda;H,r_\lambda),
\]
so the singular SSF equals the total resonance index [1608.04184]. Second,
\[
\xi^{(s)}(\lambda;\{H_r\})=-\mu^{(s)}(\lambda;\{H_r\}),
\]
where $\mu^{(s)}$ is the singular part of Pushnitski’s $\mu$-invariant, defined through scattering eigenphase flow and independent of the angle variable [1608.04184]. In the earlier trace-class setting, the angle-independence of $\mu^{(s)}$ and the equality
\[
\xi^{(s)}(\lambda;H_1,H_0)=-\mu^{(s)}(\lambda;H_1,H_0)\in\mathbb{Z}
\]
were established in "Singular spectral shift and Pushnitski $\mu$-invariant" [1009.3726].

For Schrödinger operators,
\[
H_0=-\Delta+V_0(x),\qquad H_r=H_0+rV,
\]
with $V_0$ bounded measurable real-valued and $V$ bounded integrable real-valued on $L^2(\mathbb{R}^\nu)$, the resolvent-comparable hypothesis holds for $\nu=1,2,3$ but not in general for $\nu\ge 4$ under these assumptions [1608.04184]. 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" [1004.1582], one studies a path
\[
A(t)=A_- - B(t),\qquad t\in\mathbb{R},
\]
with $A_-$ self-adjoint, $B'(t)(|A_-|+I)^{-1}\in\mathfrak{B}_1$ a.e., and
\[
\int_{\mathbb{R}}\|B'(t)(|A_-|+I)^{-1}\|_{\mathfrak{B}_1}\,dt<\infty
\]
[1004.1582]. These assumptions imply the existence of asymptotes $A_\pm$ with
\[
(A_+-A_-)(A_- - zI)^{-1}\in\mathfrak{B}_1,\qquad
(A_+-zI)^{-1}-(A_- - zI)^{-1}\in\mathfrak{B}_1
\]
[1004.1582].

The associated first-order operator
\[
D_A=\frac{d}{dt}+A
\]
leads to second-order operators
\[
H_1=D_A^*D_A,\qquad H_2=D_A D_A^*,
\]
and the central resolvent trace identity becomes
\[
\operatorname{Tr}_{L^2}\big((H_2-zI)^{-1}-(H_1-zI)^{-1}\big)
=
\frac{1}{2z}\operatorname{Tr}_{\mathcal H}\big(g_z(A_+)-g_z(A_-)\big),
\]
with
\[
g_z(x)=x(x-z)^{-1/2}
\]
[1004.1582]. This is the core Krein-type relation from which the paper derives a Pushnitski-type transform linking the SSF for $(H_2,H_1)$ to the SSF for $(A_+,A_-)$:
\[
\xi(\lambda;H_2,H_1)
=
\frac{1}{\pi}\int_{-\sqrt{\lambda}}^{\sqrt{\lambda}}
\frac{\xi(\nu;A_+,A_-)}{\sqrt{\lambda-\nu^2}}\,d\nu,
\qquad \text{a.e. }\lambda>0
\]
[1004.1582].

This framework has direct index-theoretic consequences. If $0\in\rho(A_+)\cap\rho(A_-)$, then $D_A$ is Fredholm and
\[
\operatorname{ind}(D_A)=\xi(0^+;H_2,H_1)=\xi(0;A_+,A_-)
\]
[1004.1582]. Moreover,
\[
\operatorname{ind}(D_A)=\operatorname{SpFlow}(\{A(t)\}_{t\in\mathbb{R}})=\xi(0;A_+,A_-),
\]
and also
\[
\operatorname{ind}(D_A)
=
\operatorname{Tr}_{\mathcal H}\big(E_{A_-}((-\infty,0))-E_{A_+}((-\infty,0))\big)
\]
[1004.1582].

The survey "The Spectral shift function and the Witten index" [1505.04895] places these formulas into a broader Krein-type context. It reiterates that when $0\in\rho(A_\pm)$,
\[
\operatorname{ind}(D_A)=\xi(0;A_+,A_-),
\]
and shows that in the non-Fredholm case, provided $0$ is a right and left Lebesgue point of $\xi(\cdot;A_+,A_-)$, the resolvent-regularized Witten index is
\[
W_r(D_A)
=
\xi(0_+;|D_A^*|^2,|D_A|^2)
=
\frac{\xi(0_+;A_+,A_-)+\xi(0_-;A_+,A_-)}{2}
\]
[1505.04895]. 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" [2211.14970] derives an explicit rank-two Krein-type resolvent identity. For the finite-interval operators $H_{\ell,\varphi,R}$ and Dirichlet reference operators $H_{\ell,D}$,
\[
(H_{\ell,\varphi,R}-zI)^{-1}
=
(H_{\ell,D}-zI)^{-1}+P_{\ell,\varphi,R}(z),
\]
where $P_{\ell,\varphi,R}(z)$ is rank two and is written using solutions $\psi_{\ell,1},\psi_{\ell,2}$ and a $2\times 2$ matrix $K_{\ell,\varphi,R}(z)$ [2211.14970]. The paper interprets this in boundary-triplet form as
\[
(A_\Theta-z)^{-1}=(A_0-z)^{-1}+\gamma(z)(\Theta-M(z))^{-1}\gamma(\bar z)^*,
\]
with $K_{\ell,\varphi,R}(z)=\Theta-M(z)$ [2211.14970]. In this setting, the Birman–Krein scattering formula is not used; the analysis proceeds through resolvent identities and trace-ideal control alone [2211.14970].

For singular Sturm–Liouville operators, "Explicit Krein Resolvent Identities for Singular Sturm-Liouville Operators with Applications to Bessel Operators" [1908.05392] gives rank-one and rank-two formulas in terms of boundary condition bases and the Lagrange bracket. In the one-limit-circle case,
\[
(T_\theta-zI)^{-1}-(T_0-zI)^{-1}
=
k_\theta(z)^{-1}(w_z,\cdot)\,w_z,
\]
with
\[
k_\theta(z)=\cot(\theta)+[w_z,\psi_a](a),
\]
so the resolvent difference is rank one [1908.05392]. Consequently,
\[
\operatorname{tr}\big((T_\theta-zI)^{-1}-(T_0-zI)^{-1}\big)
=
\frac{(w_z,w_z)}{\cot(\theta)+[w_z,\psi_a](a)}
\]
[1908.05392]. In the Bessel application, this leads to an explicit perturbation determinant $m_{\theta,\nu}(z)$ and
\[
\frac{d}{dz}\ln m_{\theta,\nu}(z)
=
\operatorname{tr}\big((T_\theta^{(\nu)}-zI)^{-1}-(T_F^{(\nu)}-zI)^{-1}\big),
\]
followed by the spectral shift representation
\[
\xi(\lambda;T_\theta^{(\nu)},T_F^{(\nu)})
=
\lim_{\varepsilon\downarrow0}\Big[-\frac{1}{\pi}\arg m_{\theta,\nu}(\lambda+i\varepsilon)\Big]+c_{\theta,\nu}
\]
[1908.05392].

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" [1011.5034], the Weyl function is a boundary $S$-matrix $S(\lambda)$ rather than a scattering matrix in the usual asymptotic sense. For a regular self-adjoint extension $A_L$ relative to the Friedrichs extension $A_F$, the paper obtains
\[
\operatorname{Tr}\big((A_L-\lambda)^{-1}-(A_F-\lambda)^{-1}\big)
=
\frac{D'(\lambda)}{D(\lambda)},
\qquad
D(\lambda)=\det(P+Q S(\lambda)),
\]
and writes
\[
D(\lambda)=\exp(-2\pi i\,\widetilde\xi(\lambda))
\]
[1011.5034]. The associated step-function spectral shift has jumps equal to eigenvalue multiplicity differences [1011.5034]. The same determinant controls zeta-regularized determinant comparisons:
\[
\det_\zeta(A_L-\widetilde\lambda)
=
e^{-T}D(\widetilde\lambda)\det_\zeta(A_F-\widetilde\lambda)
\]
[1011.5034]. 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" [1111.4377] defines the SSF by an invariance principle from semigroups and proves
\[
\operatorname{Tr}\big[J\,g(H_Y)\,J^*-g(H)\big]
=
\int_0^\infty g'(\lambda)\,\xi(\lambda)\,d\lambda
\]
for a specific class of test functions [1111.4377]. The Birman–Krein formula takes the form
\[
e^{-2\pi i\,\xi(\lambda)}=\det S(\lambda)\qquad\text{for a.e. }\lambda>0
\]
[1111.4377]. In two dimensions the threshold behavior is logarithmic:
\[
\xi(\lambda)
=
\frac{1}{-\log\lambda}
+
\frac{C(K)-\log 4+2\gamma}{(-\log\lambda)^2}
+
\frac{(C(K)-\log 4+2\gamma)^2-\pi^2/3}{(-\log\lambda)^3}
+
o((-\log\lambda)^{-3}),
\qquad \lambda\downarrow0,
\]
with $C(K)=-4\pi R(K)$ [1111.4377]. This low-energy expansion is then transferred to large-time asymptotics of the pinned Wiener sausage [1111.4377].

For obstacle assemblies, "A relative trace formula for obstacle scattering" [2002.07291] introduces a relative spectral shift
\[
\xi_{\mathrm{rel}}(\lambda)
=
\xi_{\mathcal O}(\lambda)-\sum_{j=1}^N \xi_{\mathcal O_j}(\lambda)
=
\frac{1}{2\pi i}\log\frac{\det S(\lambda)}{\prod_{j=1}^N\det S_j(\lambda)},
\]
so interior eigenvalue terms cancel [2002.07291]. The corresponding relative trace is encoded by a holomorphic function $\Xi$ satisfying
\[
\frac{1}{\pi}\operatorname{Im}\Xi(\lambda)=-\xi_{\mathrm{rel}}(\lambda)
\]
on the positive real axis, and
\[
\operatorname{Tr}(D_f)=\frac{i}{2\pi}\int_{\widetilde\Gamma_\varepsilon}\Xi(\lambda)f'(\lambda)\,d\lambda
\]
for a large sectorial class of test functions [2002.07291]. The interaction part of the shift is represented by a boundary-layer determinant
\[
\Xi(\lambda)=\log\det(Q_\lambda \widetilde Q_\lambda^{-1})
\]
[2002.07291]. 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" [2503.19734] 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 [2503.19734]. The paper emphasizes that for periodic backgrounds one typically works with per-period traces or relative normalizations because $H-H_0$ is usually not trace class globally [2503.19734].

For first-order ODEs with $S$-periodic boundary conditions, "Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs" [1504.01815] uses conditional Fredholm determinants rather than an SSF on the real line. The central identity compares
\[
\det\big((d/dt-D+\nu I_n)(d/dt+\widehat P_0)^{-1}\big)
\]
to the monodromy determinant $\det(S\gamma_D(T)-e^{\nu T}I_n)$ [1504.01815]. Differentiation yields trace formulas for powers of
\[
F=D(d/dt-D_0+\nu I_n)^{-1}
\]
that the paper explicitly describes as a non-self-adjoint analogue of the Hamiltonian case [1504.01815]. 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" [2102.00090] considers self-adjoint $H$ and bounded self-adjoint $V$ with
\[
V(H-iI)^{-1}\in\mathcal{S}^n.
\]
It proves the existence of a real-valued higher-order spectral shift function $\xi_n$ such that for $f$ in a concrete class $W_n$,
\[
\operatorname{Tr}\Big(
f(H+V)-\sum_{k=0}^{n-1}\frac{1}{k!}\frac{d^k}{dt^k}f(H+tV)\big|_{t=0}
\Big)
=
\int_{\mathbb{R}} f^{(n)}(\lambda)\,\xi_n(\lambda)\,d\lambda
\]
[2102.00090]. The function is unique up to a polynomial of degree at most $n-1$ and satisfies weighted integrability bounds [2102.00090]. 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" [1805.01337] defines a spectral shift function for pairs of nonpositive or negative operators under nuclear perturbations. If $A,B\in N(X)$ and $A-B\in G_1(X)$, then for negative complete Bernstein functions $\varphi$ satisfying the paper’s integrability assumptions,
\[
\operatorname{tr}(\varphi(A)-\varphi(B))
=
\frac{1}{2\pi i}\oint_{\partial\Omega_{A,B}}\xi_{A,B}(z)\,\varphi'(z)\,dz,
\]
where
\[
\xi_{A,B}(\lambda)=\operatorname{tr}\big(\log(\lambda I-A)-\log(\lambda I-B)\big)
\]
for $\lambda>0$ and is analytically continued to a sector [1805.01337]. 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" [2603.21773] defines the SSF as a distribution via
\[
(\xi',f)=\operatorname{Tr}(f(H)-f(H_0))
\]
for $f\in\mathcal D(I)$ under hypotheses controlling non-real eigenvalues, resolvent growth, and relatively trace-class behavior [2603.21773]. The key jump formula is
\[
\xi'(\cdot;H,H_0)
=
\frac{1}{2\pi i}\lim_{\varepsilon\to0+}
\big(o(\cdot+i\varepsilon)-o(\cdot-i\varepsilon)\big),
\qquad
o(z)=\operatorname{Tr}(R_H(z)-R_{H_0}(z)),
\]
in the sense of distributions [2603.21773]. The determinant representation becomes
\[
\xi(\lambda;H,H_0)
=
\frac{1}{2\pi i}\lim_{\varepsilon\to0+}
\big(\ln D_V(\lambda+i\varepsilon)-\ln D_V(\lambda-i\varepsilon)\big),
\]
with $D_V(z)=\det(I+V(H_0-z)^{-1})$ [2603.21773]. Here $\xi$ may be complex-valued, spectral singularities produce principal-value and delta-derivative terms in $\xi'$, and the classical unitary Birman–Krein identity is no longer expected in unchanged form [2603.21773].

A parallel non-self-adjoint direction concerns contractions and dissipative operators. "Real-valued spectral shift functions for contractions and dissipative operators" [2410.22529] and its expanded version [2603.25242] show that for contractions $T_0,T_1$ with $T_1-T_0\in S_1$, there exists an integrable SSF $\xi$ on $\mathbb T$ such that
\[
\operatorname{trace}\big(\varphi(T_1)-\varphi(T_0)\big)
=
\int_{\mathbb T}\varphi'(\zeta)\,\xi(\zeta)\,dm(\zeta)
\]
for $\varphi$ in the operator-Lipschitz disk-algebra class [2410.22529], [2603.25242]. Unlike the self-adjoint case, $\xi$ is nonunique: adding any $H^1(\mathbb T)$ function yields another SSF [2410.22529], [2603.25242]. Real-valued representatives require extra structure. For maximal dissipative operators one has the rational-function trace formula
\[
\operatorname{trace}\big(f(L_1)-f(L_0)\big)=\int_{\mathbb R} f'(t)\,\xi(t)\,dt
\]
under trace-class or resolvent-difference hypotheses, but a real-valued integrable SSF can fail to exist unless additional conditions hold; in particular, if $L_1-L_0\in S_1$ and $\operatorname{trace}(L_1-L_0)\notin\mathbb R$, then no real-valued integrable SSF exists [2603.25242].

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 $\xi$, the ambient functional calculus, and the status of real-valuedness, absolute continuity, and scattering interpretation [1608.04184], [1004.1582], [2102.00090], [2603.21773], [2603.25242].

Source: https://www.emergentmind.com/topics/krein-type-spectral-shift-formula