Boundary Regularized Fredholm Determinant

• Boundary Regularized Fredholm Determinant is a modification that compresses bulk spectral invariants to boundary operators, enabling finite-dimensional analysis.
• It leverages Dirichlet-to-Neumann maps and boundary trace operators to transform infinite-dimensional determinants into more manageable boundary determinants.
• This methodology enhances both analytical and numerical spectral analysis, applying to multi-dimensional, nonlocal, and index-related problems in spectral geometry.

A boundary regularized Fredholm determinant is a modification of the classical Fredholm determinant construction in which the spectral, scattering, or trace invariants of operators acting in a bulk domain (typically on a full Hilbert space such as $L^2(\Omega)$ or a Sobolev space inside a manifold) are “compressed” or “reduced” to operators acting only on the boundary. This is achieved via mapping constructs such as Dirichlet-to-Neumann (DtN) operators, boundary data maps, or related boundary trace operators. The central technical advantage is that infinite-dimensional determinants (often requiring nontrivial regularization in function spaces) are replaced or regularized by finite- or lower-dimensional determinants on boundary spaces (e.g., $L^2(\partial\Omega)$). This approach extends and generalizes foundational spectral reduction results, such as the Jost–Pais formula, to multi-dimensional, nonlocal, and even non-selfadjoint settings.

1. Boundary Determinant Reduction via Dirichlet-to-Neumann Maps

Boundary regularization of Fredholm determinants primarily leverages the Dirichlet-to-Neumann map, defined for Schrödinger-type operators with suitable boundary conditions and acting in $L^2(\Omega)$ for $\Omega\subset\mathbb{R}^n$ with regular boundary. For a Schrödinger operator $H=-\Delta+V$ in $\Omega$, the DtN maps for the perturbed and unperturbed problems—denoted $M_\Omega(z)$ and $M_{0,\Omega}(z)$—yield the central reduction formula: $$\det_2\Bigl(I_{L^2(\Omega)} + u(H^D_\Omega - zI)^{-1}v\Bigr) \, \det_2\Bigl(I_{L^2(\Omega)} + u(H^N_\Omega - zI)^{-1}v\Bigr)^{-1} = \det_2\Bigl(M_\Omega(z) M_{0,\Omega}(z)^{-1}\Bigr) \, \exp\bigl(\operatorname{tr}(T_2(z))\bigr)$$ (see (Gesztesy et al., 2010), Equations (1.18)-(1.19)). Here, $u$, $v$ are square-root factorizations of the potential ($V=uv$), and $T_2(z)$ is an explicit correction operator. This reduces the ratio of bulk Fredholm determinants to a determinant involving only boundary operators acting on $L^2(\partial\Omega)$.

The DtN map itself encodes the entire bulk geometry and potential information (local or nonlocal), transforming Dirichlet boundary data into Neumann data and serving as the critical object in the boundary reduction. Spectral properties, such as the eigenvalue set $\sigma(H_\Omega)$, are equivalent to the vanishing of the boundary regularized determinant via the relation: $$z \in \sigma(H_\Omega) \quad \Longleftrightarrow \quad \det_2(I - \Gamma(z)) = 0$$ where $\Gamma(z) = M_\Omega(z)M_{0,\Omega}(z)^{-1}$.

2. Extension of the Jost–Pais Formula and Nonlocal Interactions

The classical Jost–Pais formula establishes that in one dimension, the Fredholm determinant for a half-line Schrödinger operator reduces to a simple Wronskian of the Jost solution. Boundary regularization generalizes this concept to higher dimensions and nonlocal potentials. For Schrödinger operators with nonlocal interactions (factored $V=vu$ with Hilbert–Schmidt $u$, $v$), the modified determinant formula becomes: $$\det(I + u (H_{D, \Omega} - zI)^{-1} v) \cdot \det(I + u (H_{N, \Omega} - zI )^{-1} v) = \det(\widetilde{M}(z) M_{0, 2}(z)^{-1})$$ ((Gesztesy et al., 2010), Equation (4.23)), where $\widetilde{M}(z)$ and $M_{0,2}(z)$ are the DtN maps for the perturbed and reference problems, respectively. The left-hand side involves bulk determinants in $L^2(\Omega)$, while the right-hand side is solely a boundary determinant. This compression persists in the nonlocal setting, thus encompassing a broad class of Schrödinger operators previously inaccessible to purely local analytic techniques.

3. Symmetrized Boundary Data Maps and Finite-Dimensional Determinants

In one-dimensional settings, boundary regularization can yield explicit finite-dimensional determinants. For Schrödinger operators on a compact interval $[0,R]$ with separated or mixed boundary conditions, there is an abstract reduction for symmetrized Fredholm determinants: $$\det\left[(A-zI)^{1/2} (A_0-zI)^{-1} (A-zI)^{1/2}\right] \propto \det(A_{0,0}^{R}(z))$$ (Gesztesy et al., 2010), where $A_{0,0}^{R}(z)$ is a $2 \times 2$ boundary data map explicitly constructed from traces of solution values and derivatives at the boundary. This provides concrete trace formulas for resolvent differences and spectral shift functions, recasting infinite dimensional determinants as explicit finite objects fully determined by boundary data.

4. Integral Kernel Factorizations and Boundary Regularization

Reduction to boundary determinants is further enabled by the structure of integral kernels with semi-separable or Volterra-type factorizations, as described in (Carey et al., 2014). For an operator $\mathbf{K}$ in $L^2((a,b);\mathcal{H})$ with semi-separable kernel

$$K(x,x') = F_1(x) G_1(x') \ \text{for} \ x' < x, \qquad K(x,x') = F_2(x) G_2(x') \ \text{for} \ x' > x$$

the modified Fredholm determinant admits the factorization

$$\det_2(I - aK) = \det_{\mathcal{H}_1}(I - QT(I - aH_b)^{-1}S) = \det_{\mathcal{H}_2}(I - R(I - aH_a)^{-1}Q)$$

(see Theorem 2.13). This reduces the bulk Fredholm determinant to a boundary determinant in $\mathcal{H}_1$ or $\mathcal{H}_2$, where $Q$, $T$, $S$, and $R$ are finite-rank linking operators canonically associated to the boundary structure.

Such reductions underpin computations of transmission coefficients, index formulas, and trace formulas: for operator-valued Schrödinger potentials, this translates to expressing spectral invariants exclusively in terms of boundary Jost functions or related boundary objects.

5. Boundary Regularization in Spectral, Index, and Zeta Determinants

The boundary regularization paradigm is ubiquitous in spectral theory and index theory, especially in the definition of trace, index, and determinant invariants for operators with boundary conditions. For example, in the Lorentzian Dirac operator setting (Baer et al., 2017) or in elliptic first-order boundary problems (Baer et al., 2019), the imposition of “elliptic” or generalized boundary conditions (using spectral projections or graph-type conditions) ensures the operator is Fredholm. The index of the boundary regularized realization

$$\operatorname{ind}(D_B) = \dim \ker D_B - \dim \ker D_B^*$$

is determined by the choice of boundary condition, often via a K-theory class, reflecting the topological corrections arising from boundary contributions (Prokhorova, 29 Jan 2024).

Boundary regularization also interfaces with zeta-regularized determinants in global analysis (Hartmann et al., 2021). There, the regularized Fredholm determinant, $\det_p(I + z L^{-1})$, is related to the zeta determinant via explicit heat trace invariants and correction terms: $$\frac{\det_{\zeta}(L+z)}{\det_{\zeta}(L)} = \exp\left(\sum_{j=1}^{p-1} \frac{z^j}{j!} \left.\frac{d^j}{dz^j}\log\det_{\zeta}(L+z)\right|_{z=0}\right) \cdot \det_p(I + z L^{-1})$$ with the asymptotics precisely controlled by boundary spectral invariants (heat trace coefficients).

6. Product Formulas and Boundary Correction Terms

Boundary regularized determinants also modify multiplicativity properties of Fredholm determinants, necessitating explicit correction terms for products of regularized determinants. For higher Schatten class operators (e.g., Hilbert–Schmidt), the product formula is: $$\det_2((I - A)(I - B)) = \det_2(I - A) \det_2(I - B) \exp(-\operatorname{tr}(AB))$$ (Britz et al., 2020). For $k$-regularized determinants, the correction term generalizes to $\exp(\operatorname{tr}(X_{H,k}(A,B)))$, where $X_{H,k}(A,B)$ is a computable polynomial in $A$, $B$ of order at least $k$. The exponent encodes “boundary” interactions between operators akin to interface or cutoff effects often encountered in spectral problems with boundary conditions (Koutsonikos-Kouloumpis et al., 2022).

7. Numerical Applications and Robustness

Boundary regularized Fredholm determinant methods play a central role in accurate large-scale numerical computations of spectral quantities. In the boundary integral formulation for the Laplacian (the “drum problem”), the Fredholm determinant for the double-layer potential operator is discretized using spectrally accurate Nyström quadrature. The roots of the determinant correspond to Dirichlet eigenvalues of the Laplacian, and exponential convergence of the numerical determinant with respect to the quadrature points is established for analytic boundaries (Zhao et al., 2014).

Combined field representations and analytic root-finding algorithms (e.g., Boyd’s method) further alleviate numerical instabilities associated with spurious boundary resonances, ensuring high accuracy and computational efficiency even for non-simply-connected domains.

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Boundary regularized Fredholm determinants enable the reduction of spectral, index, and trace invariants from high-dimensional operators to boundary objects, thereby facilitating both analytical and numerical computation. This methodology provides a versatile and mathematically rigorous bridge between infinite-dimensional operator theory, boundary value problems, and spectral geometry, encompassing applications from multi-dimensional scattering theory to spectral shift and index theorems, and powering robust algorithms in computational spectral analysis.