Jacobi-Steklov Spectrum Overview

• Jacobi-Steklov spectrum is defined as a boundary eigenvalue problem where the spectral parameter appears in the boundary condition rather than the interior, extending classical Steklov theory with Jacobi operators.
• It employs a first-order elliptic self-adjoint pseudodifferential framework that yields discrete, real, and unbounded eigenvalues governed by Weyl-type asymptotics with subprincipal corrections.
• Applications span free-boundary minimal surface analysis to magnetic and electromagnetic problems, providing insights into boundary geometry and spectral invariants.

The Jacobi–Steklov spectrum is the spectrum of a boundary spectral problem in which the spectral parameter appears in the boundary condition rather than in the interior equation. In its standard form, one considers a second-order elliptic operator $J$, often a Jacobi operator such as $J=\Delta+q(x)$, and a Steklov-type boundary condition such as $\partial_\nu u+\beta(x)u=\sigma u$. The classical Steklov problem for the Laplacian,

$$\Delta u=0 \text{ in } \Omega,\qquad \partial_\nu u=\sigma u \text{ on } \partial\Omega,$$

is the canonical simplest example. In smooth settings the associated boundary operator is a first-order elliptic self-adjoint pseudodifferential operator, so the spectrum is discrete, real, and unbounded, and it is naturally interpreted as the spectrum of a generalized Dirichlet-to-Neumann map (Girouard et al., 2014, Chakradhar et al., 2024).

1. Definition and operator-theoretic formulation

For a smooth compact manifold with boundary $M$, the classical Dirichlet-to-Neumann operator is defined by harmonic extension: if $f$ is prescribed on $\partial M$, one solves

$$\Delta(Hf)=0 \text{ in } M,\qquad Hf|_{\partial M}=f,$$

and sets

$\Lambda f=\partial_\nu(Hf).$

The Steklov eigenvalue problem is then $\Lambda f=\sigma f$, equivalently

$J=\Delta+q(x)$0

Its spectrum is discrete and nonnegative,

$J=\Delta+q(x)$1

under the regularity assumptions used in the standard Steklov theory (Girouard et al., 2014).

The Jacobi–Steklov generalization replaces the interior Laplacian by a Jacobi-type operator. In the supplied formulations, one natural model is

$J=\Delta+q(x)$2

with boundary condition

$J=\Delta+q(x)$3

More generally, one may consider

$J=\Delta+q(x)$4

solve $J=\Delta+q(x)$5 with prescribed boundary values, and define the generalized Dirichlet-to-Neumann map $J=\Delta+q(x)$6 by $J=\Delta+q(x)$7, where $J=\Delta+q(x)$8 is the boundary operator appearing in the Steklov-type condition. The resulting eigenvalues are the Jacobi–Steklov eigenvalues (Girouard et al., 2013, Girouard et al., 2014).

Several standard generalizations already fit this scheme. A weighted Steklov problem is

$J=\Delta+q(x)$9

with boundary density $\partial_\nu u+\beta(x)u=\sigma u$0, and a magnetic Steklov problem is

$\partial_\nu u+\beta(x)u=\sigma u$1

where $\partial_\nu u+\beta(x)u=\sigma u$2 is the magnetic Laplacian and $\partial_\nu u+\beta(x)u=\sigma u$3 is the magnetic Steklov operator (Colbois et al., 2011, Chakradhar et al., 2024).

2. Pseudodifferential structure and high-energy asymptotics

For smooth boundary, the Dirichlet-to-Neumann operator is a first-order elliptic self-adjoint pseudodifferential operator whose principal symbol is $\partial_\nu u+\beta(x)u=\sigma u$4 with respect to the induced boundary metric. This immediately places Steklov and Jacobi–Steklov problems in the standard microlocal framework for elliptic $\partial_\nu u+\beta(x)u=\sigma u$5DOs and yields Weyl-type asymptotics for the counting function (Girouard et al., 2013, Girouard et al., 2014).

In dimension $\partial_\nu u+\beta(x)u=\sigma u$6, the classical Steklov counting function satisfies

$\partial_\nu u+\beta(x)u=\sigma u$7

and equivalently

$\partial_\nu u+\beta(x)u=\sigma u$8

On surfaces this simplifies drastically. If the boundary is connected and has length $\partial_\nu u+\beta(x)u=\sigma u$9, then

$$\Delta u=0 \text{ in } \Omega,\qquad \partial_\nu u=\sigma u \text{ on } \partial\Omega,$$0

and for multiple boundary components the asymptotic model is a rearranged union of arithmetic progressions $$\Delta u=0 \text{ in } \Omega,\qquad \partial_\nu u=\sigma u \text{ on } \partial\Omega,$$1, each appearing twice (Girouard et al., 2013).

For Jacobi–Steklov operators, the same principal-symbol argument implies that lower-order terms do not alter the order of growth. A standard consequence is that the leading Weyl term is governed by the boundary measure, while lower-order data enter only in subprincipal corrections. In the supplied formulations this is repeatedly expressed by saying that a generalized Dirichlet-to-Neumann operator for $$\Delta u=0 \text{ in } \Omega,\qquad \partial_\nu u=\sigma u \text{ on } \partial\Omega,$$2 remains an elliptic first-order operator with the same principal symbol structure; this suggests that the leading Weyl growth matches the classical Steklov law (Girouard et al., 2014, Girouard et al., 2013).

A model beyond the smooth case is provided by spherical cylinders $$\Delta u=0 \text{ in } \Omega,\qquad \partial_\nu u=\sigma u \text{ on } \partial\Omega,$$3. For these domains, the Steklov counting function admits a two-term asymptotic expansion in which the second term contains both a curvature contribution from smooth boundary pieces and an edge contribution from the codimension-two strata where the lateral boundary meets the bases. The edge coefficient is represented by explicit angular and Beta-function integrals, for example

$$\Delta u=0 \text{ in } \Omega,\qquad \partial_\nu u=\sigma u \text{ on } \partial\Omega,$$4

which is a universal right-angle edge constant in the second term (Bullent, 25 Jun 2025).

3. Spectral invariants of boundary geometry

A central feature of Steklov-type spectra is that they are strongly boundary-sensitive. For smooth compact surfaces, the Steklov spectrum determines the number of boundary components and the lengths of those components. In particular, if $$\Delta u=0 \text{ in } \Omega,\qquad \partial_\nu u=\sigma u \text{ on } \partial\Omega,$$5 is the length of the largest boundary component, then

$$\Delta u=0 \text{ in } \Omega,\qquad \partial_\nu u=\sigma u \text{ on } \partial\Omega,$$6

This is obtained from the refined asymptotic model $$\Delta u=0 \text{ in } \Omega,\qquad \partial_\nu u=\sigma u \text{ on } \partial\Omega,$$7, where $$\Delta u=0 \text{ in } \Omega,\qquad \partial_\nu u=\sigma u \text{ on } \partial\Omega,$$8 and $$\Delta u=0 \text{ in } \Omega,\qquad \partial_\nu u=\sigma u \text{ on } \partial\Omega,$$9 is the reordered union of the corresponding arithmetic progressions (Girouard et al., 2013).

For compact Riemannian orbisurfaces, the boundary components come in two types: smooth circles and type II components, which are quotient circles with two reflector singularities. The Steklov spectrum determines the numbers of type I and type II boundary components and therefore detects the presence and number of orbifold singularities on the boundary. It also determines the boundary lengths modulo the explicit equivalence relation

$M$0

which is the precise ambiguity in the two-dimensional orbifold setting (Arias-Marco et al., 2016).

At the same time, the Steklov spectrum does not detect all singular geometric features. A flat disk can be Steklov isospectral to a cone: for the flat cone $M$1 obtained as a quotient of a disk by a rotation of angle $M$2, the Dirichlet-to-Neumann operator coincides with that of the flat disk $M$3. Consequently, the Steklov spectrum does not detect interior orbifold singularities and does not determine the orbifold Euler characteristic (Arias-Marco et al., 2016).

For convex polygons, the characteristic polynomial $M$4 extracted from the Steklov spectrum encodes side lengths and angles through trigonometric frequencies and angle-dependent coefficients. Within this framework, almost all triangles are uniquely determined by their Steklov spectra within the class of all triangles; rectangles are uniquely determined among all triangles and convex quadrilaterals; and triangles and quadrilaterals are spectrally distinguished from simply connected smooth domains. In this sense, corners are spectrally visible in low polygonal complexity (Dryden et al., 21 Apr 2026).

4. Nonsmooth domains, edges, and singular spectral phenomena

Nonsmooth boundary geometry changes the subleading structure of the spectrum. In piecewise smooth domains, the principal Weyl term remains determined by total boundary measure, but the next term collects contributions from smooth curvature and from edges. Spherical cylinders $M$5 are a model example: the lateral boundary contributes curvature, the flat bases contribute differently, and the right-angle intersections produce explicit edge terms weighted by universal constants such as $M$6 (Bullent, 25 Jun 2025).

Geometric singularities can also change the spectral type itself. In a bounded domain with a cusp of sharpness exponent $M$7, the continuous spectrum of the Steklov problem is

$M$8

If the cusp is blunted at scale $M$9, the spectrum on the truncated domains is discrete, but as $f$0 three kinds of behavior appear: stable eigenvalues, blinking eigenvalues, and gliding eigenvalues. Blinking eigenvalues reappear near any fixed $f$1 along logarithmically spaced sequences

$f$2

and this is part of the mechanism by which the discrete spectra of the blunted domains generate the continuous spectrum of the cuspidal limit (Nazarov et al., 2018).

Another singular phenomenon concerns perturbations of rough boundaries. Under Lipschitz graph convergence together with perimeter convergence, the Steklov resolvents converge compactly in the sense of Vainikko, so each eigenvalue converges and eigenfunctions converge strongly in $f$3 after identification by the connecting system. If the surface elements converge only weakly, the limit problem becomes a weighted Steklov problem

$f$4

and if the boundary oscillates too strongly, all positive eigenvalues collapse to $f$5. In periodically corrugated domains $f$6, one obtains a trichotomy: stability for $f$7, a weighted limit for $f$8, and degeneration for $f$9 (Ferrero et al., 2021).

5. Bounds, comparison principles, and coarse control

A large part of the modern theory concerns upper and lower bounds on normalized Steklov eigenvalues. For a bounded domain $\partial M$0 with boundary $\partial M$1, the normalized eigenvalues are

$\partial M$2

and, in the weighted setting,

$\partial M$3

If the ambient metric is conformally equivalent to a complete metric with nonnegative Ricci curvature, then

$\partial M$4

For bounded domains in $\partial M$5, $\partial M$6, or a hemisphere of $\partial M$7, this yields the uniform bound

$\partial M$8

On compact orientable surfaces of genus $\partial M$9,

$$\Delta(Hf)=0 \text{ in } M,\qquad Hf|_{\partial M}=f,$$0

These inequalities are not Weyl-sharp in the exponent of $$\Delta(Hf)=0 \text{ in } M,\qquad Hf|_{\partial M}=f,$$1, but they are uniform and geometric (Colbois et al., 2011).

There are also direct comparison inequalities between Steklov eigenvalues and boundary Laplace eigenvalues. If $$\Delta(Hf)=0 \text{ in } M,\qquad Hf|_{\partial M}=f,$$2 are the Laplace–Beltrami eigenvalues of the boundary and

$$\Delta(Hf)=0 \text{ in } M,\qquad Hf|_{\partial M}=f,$$3

then, under nonnegative Ricci curvature and a mild localization hypothesis,

$$\Delta(Hf)=0 \text{ in } M,\qquad Hf|_{\partial M}=f,$$4

This makes precise the principle that large boundary Laplace eigenvalues constrain normalized Steklov eigenvalues (Colbois et al., 2011).

For manifolds with cylindrical boundary and bounded geometry, Steklov eigenvalues can also be compared to those of coarse discretizations. An $$\Delta(Hf)=0 \text{ in } M,\qquad Hf|_{\partial M}=f,$$5-discretization produces a graph with boundary $$\Delta(Hf)=0 \text{ in } M,\qquad Hf|_{\partial M}=f,$$6 roughly isometric to the manifold, and there exist constants $$\Delta(Hf)=0 \text{ in } M,\qquad Hf|_{\partial M}=f,$$7, depending only on the geometry class and $$\Delta(Hf)=0 \text{ in } M,\qquad Hf|_{\partial M}=f,$$8, such that

$$\Delta(Hf)=0 \text{ in } M,\qquad Hf|_{\partial M}=f,$$9

with analogous $\Lambda f=\partial_\nu(Hf).$0-dependent estimates for higher eigenvalues. This framework is used to build graph-like surfaces with fixed boundary length and arbitrarily large Steklov spectral gap, including examples with connected boundary (Colbois et al., 2016).

6. Generalizations: magnetic, electromagnetic, and geometric Jacobi–Steklov models

The magnetic Steklov operator is a direct analogue of a Jacobi–Steklov operator with lower-order structure. For a magnetic potential $\Lambda f=\partial_\nu(Hf).$1, one defines

$\Lambda f=\partial_\nu(Hf).$2

and, for boundary data $\Lambda f=\partial_\nu(Hf).$3, the $\Lambda f=\partial_\nu(Hf).$4-harmonic extension $\Lambda f=\partial_\nu(Hf).$5 solves $\Lambda f=\partial_\nu(Hf).$6. The magnetic Steklov operator is

$\Lambda f=\partial_\nu(Hf).$7

Its first eigenvalue satisfies

$\Lambda f=\partial_\nu(Hf).$8

Moreover,

$\Lambda f=\partial_\nu(Hf).$9

equivalently $\Lambda f=\sigma f$0 and $\Lambda f=\sigma f$1 for every closed curve $\Lambda f=\sigma f$2. In that case $\Lambda f=\sigma f$3 is unitarily equivalent to the classical Steklov operator. The magnetic theory also includes a Cheeger–Jammes type lower bound

$\Lambda f=\sigma f$4

and comparison with the boundary magnetic Laplacian: $\Lambda f=\sigma f$5 under the curvature and collar assumptions stated in the paper (Chakradhar et al., 2024).

A vector-valued Steklov analogue appears for time-harmonic Maxwell equations. In a cavity $\Lambda f=\sigma f$6, one studies

$\Lambda f=\sigma f$7

with boundary condition of Steklov type,

$\Lambda f=\sigma f$8

After a coercive modification by a penalty term $\Lambda f=\sigma f$9, one obtains a compact self-adjoint operator on

$J=\Delta+q(x)$00

a min–max characterization,

$J=\Delta+q(x)$01

and, in the unit ball, two explicit eigenvalue families obtained from vector spherical harmonics,

$J=\Delta+q(x)$02

together with a second family depending on $J=\Delta+q(x)$03. Both satisfy $J=\Delta+q(x)$04 and $J=\Delta+q(x)$05 as $J=\Delta+q(x)$06, matching the first-order boundary character of Steklov-type operators (Ferraresso et al., 2022).

The geometric motivation for the phrase “Jacobi–Steklov” is especially strong in minimal and free-boundary geometry. A free boundary submanifold in the unit ball is characterized by the fact that the coordinate functions satisfy

$J=\Delta+q(x)$07

so each coordinate function is a Steklov eigenfunction with eigenvalue $J=\Delta+q(x)$08. In that setting, replacing $J=\Delta+q(x)$09 by the Jacobi operator of the minimal or constant-mean-curvature hypersurface produces the natural Jacobi–Steklov analogue, with the same boundary-operator viewpoint and the same expectation of first-order pseudodifferential asymptotics (Girouard et al., 2014).