Gelfand–Levitan Condition in Inverse Spectral Theory

• Gelfand–Levitan condition is an admissibility criterion ensuring that inverse spectral or scattering data generate a solvable integral equation yielding a physical potential.
• It encompasses classical formulations, Boundary Control reformulations, and Sturm–Liouville sum rules that tie spectral data to boundary parameters.
• Numerical implementations enforce related data-range, truncation, decay, and stability conditions to ensure that reconstruction methods are practical and reliable.

The expression Gelfand–Levitan condition does not denote a single universally fixed object across the literature. In the sources considered here, it refers to several closely related admissibility and compatibility requirements attached to inverse spectral and inverse scattering constructions: solvability of the classical Gelfand–Levitan equation from spectral data, positivity of an operator built from dynamical or response data, boundary-parameter compatibility identities for norming constants, and, in numerical Gelfand–Levitan–Marchenko settings, operational conditions that make reconstruction stable and computable (Avdonin et al., 29 May 2025).

1. Terminological scope and principal meanings

In the most explicit formulation among these sources, the phrase is tied to three equivalent or near-equivalent structures: the classical global Gelfand–Levitan equation, a local Gelfand–Levitan equation obtained through the Boundary Control method, and positivity/solvability conditions on operators constructed from inverse data (Avdonin et al., 29 May 2025). In this sense, the condition is an admissibility criterion: given candidate inverse data, one asks whether they correspond to a genuine potential and whether the associated integral equation is solvable.

A second usage appears in regular Sturm–Liouville inverse theory, where the Gelfand–Levitan method yields exact compatibility identities linking norming constants to separated boundary parameters. There the condition is not an operator inequality but a sum rule that admissible spectral data must satisfy (Ashrafyan et al., 2017).

A third, more operational usage arises in numerical work on the Gel'fand–Levitan–Krein and Gelfand–Levitan–Marchenko equations. Those papers explicitly state that they do not define a standalone item called the Gelfand–Levitan condition; instead they impose data-range, truncation, decay, discretization, and stability hypotheses under which the inverse problem is numerically tractable (Karchevsky et al., 2013, Medvedev et al., 2021, Medvedev et al., 2024). A plausible synthesis is that the phrase names whatever compatibility requirement ensures that the relevant GL/GLK/GLM equation corresponds to admissible inverse data.

2. Classical inverse spectral formulation

For the half-line Schrödinger operator

$H=-\partial_x^2+q(x)$

on $L^2(\mathbb R_+)$ with Dirichlet condition at $x=0$, the classical Gelfand–Levitan construction begins from the spectral measure $d\rho(\lambda)$. The regularized spectral function is

$\sigma(\lambda)= \begin{cases} \rho(\lambda)-\dfrac{2}{3\pi}\lambda^{3/2}, & \lambda\ge 0,\[1ex] \rho(\lambda), & \lambda<0, \end{cases}$

and the associated kernel is

$$F(x,t)=\int_{-\infty}^{\infty} \frac{\sin(\sqrt\lambda\,x)\sin(\sqrt\lambda\,t)}{\lambda}\,d\sigma(\lambda).$$

If $\varphi(x,\lambda)$ solves

$$-\varphi''+q(x)\varphi=\lambda\varphi,\qquad \varphi(0,\lambda)=0,\qquad \varphi'(0,\lambda)=1,$$

then the transformation operator has the representation

$$\varphi(x,\lambda)=\frac{\sin(\sqrt\lambda\,x)}{\sqrt\lambda}+\int_0^x K(x,t)\frac{\sin(\sqrt\lambda\,t)}{\sqrt\lambda}\,dt,$$

and its kernel $K(x,t)$ satisfies the classical Gelfand–Levitan equation

$L^2(\mathbb R_+)$0

The potential is recovered by

$L^2(\mathbb R_+)$1

In this classical setting, the Gelfand–Levitan condition is the requirement that the data $L^2(\mathbb R_+)$2, after regularization into $L^2(\mathbb R_+)$3, produce a solvable integral equation whose solution yields a genuine potential through the diagonal derivative formula (Avdonin et al., 29 May 2025).

The same source places this within a transformation-operator framework. The kernel $L^2(\mathbb R_+)$4 satisfies a Goursat problem,

$L^2(\mathbb R_+)$5

so the condition is simultaneously spectral, integral, and hyperbolic. That interlocking structure is central to later BC reformulations.

3. Boundary Control reformulation and operator positivity

The Boundary Control approach recasts the inverse problem through the wave equation

$L^2(\mathbb R_+)$6

with zero initial data and boundary control $L^2(\mathbb R_+)$7. The response operator is

$L^2(\mathbb R_+)$8

where $L^2(\mathbb R_+)$9 is the response function. The connecting operator $x=0$0 is defined by

$x=0$1

equivalently,

$x=0$2

Hence $x=0$3 is positive definite, bounded, and boundedly invertible. It has the form

$x=0$4

with

$x=0$5

The decisive characterization theorem states: for given $x=0$6, there exists a unique $x=0$7 such that $x=0$8 is the response function corresponding to the wave problem if and only if the operator $x=0$9 constructed from $d\rho(\lambda)$0 is positive definite. In formula form, the condition is

$d\rho(\lambda)$1

This is the clearest necessary-and-sufficient operator-theoretic formulation of a Gelfand–Levitan condition in the supplied literature (Avdonin et al., 29 May 2025).

The same paper shows that this positivity criterion is spectrally identical to the Gelfand–Levitan kernel construction. The kernel of $d\rho(\lambda)$2 admits the spectral representation

$d\rho(\lambda)$3

while the response function itself satisfies

$d\rho(\lambda)$4

Thus the admissibility of spectral data, of response data, and of the connecting operator are different presentations of the same inverse datum.

The BC factorization

$d\rho(\lambda)$5

leads to a local Gelfand–Levitan equation. If

$d\rho(\lambda)$6

then the kernels $d\rho(\lambda)$7 and $d\rho(\lambda)$8 satisfy

$d\rho(\lambda)$9

The paper states the recovery formula

$\sigma(\lambda)= \begin{cases} \rho(\lambda)-\dfrac{2}{3\pi}\lambda^{3/2}, & \lambda\ge 0,\[1ex] \rho(\lambda), & \lambda<0, \end{cases}$0

Under the identifications $\sigma(\lambda)= \begin{cases} \rho(\lambda)-\dfrac{2}{3\pi}\lambda^{3/2}, & \lambda\ge 0,\[1ex] \rho(\lambda), & \lambda<0, \end{cases}$1 and $\sigma(\lambda)= \begin{cases} \rho(\lambda)-\dfrac{2}{3\pi}\lambda^{3/2}, & \lambda\ge 0,\[1ex] \rho(\lambda), & \lambda<0, \end{cases}$2, this local BC equation is equivalent to the classical global Gelfand–Levitan equation. In that precise sense, positivity of $\sigma(\lambda)= \begin{cases} \rho(\lambda)-\dfrac{2}{3\pi}\lambda^{3/2}, & \lambda\ge 0,\[1ex] \rho(\lambda), & \lambda<0, \end{cases}$3 functions as a local, dynamical version of the Gelfand–Levitan condition (Avdonin et al., 29 May 2025).

4. Sturm–Liouville compatibility identities

For the regular self-adjoint Sturm–Liouville problem

$\sigma(\lambda)= \begin{cases} \rho(\lambda)-\dfrac{2}{3\pi}\lambda^{3/2}, & \lambda\ge 0,\[1ex] \rho(\lambda), & \lambda<0, \end{cases}$4

with separated boundary conditions

$\sigma(\lambda)= \begin{cases} \rho(\lambda)-\dfrac{2}{3\pi}\lambda^{3/2}, & \lambda\ge 0,\[1ex] \rho(\lambda), & \lambda<0, \end{cases}$5

where $\sigma(\lambda)= \begin{cases} \rho(\lambda)-\dfrac{2}{3\pi}\lambda^{3/2}, & \lambda\ge 0,\[1ex] \rho(\lambda), & \lambda<0, \end{cases}$6 and $\sigma(\lambda)= \begin{cases} \rho(\lambda)-\dfrac{2}{3\pi}\lambda^{3/2}, & \lambda\ge 0,\[1ex] \rho(\lambda), & \lambda<0, \end{cases}$7, the Gelfand–Levitan method yields exact identities constraining the normalized norming constants (Ashrafyan et al., 2017).

Let $\sigma(\lambda)= \begin{cases} \rho(\lambda)-\dfrac{2}{3\pi}\lambda^{3/2}, & \lambda\ge 0,\[1ex] \rho(\lambda), & \lambda<0, \end{cases}$8 be the left-normalized solution with

$\sigma(\lambda)= \begin{cases} \rho(\lambda)-\dfrac{2}{3\pi}\lambda^{3/2}, & \lambda\ge 0,\[1ex] \rho(\lambda), & \lambda<0, \end{cases}$9

and let $$F(x,t)=\int_{-\infty}^{\infty} \frac{\sin(\sqrt\lambda\,x)\sin(\sqrt\lambda\,t)}{\lambda}\,d\sigma(\lambda).$$0 denote the corresponding normalized norming constants. Likewise let $$F(x,t)=\int_{-\infty}^{\infty} \frac{\sin(\sqrt\lambda\,x)\sin(\sqrt\lambda\,t)}{\lambda}\,d\sigma(\lambda).$$1 be the normalized norming constants for the right-normalized solution. The central theorem gives

$$F(x,t)=\int_{-\infty}^{\infty} \frac{\sin(\sqrt\lambda\,x)\sin(\sqrt\lambda\,t)}{\lambda}\,d\sigma(\lambda).$$2

and

$$F(x,t)=\int_{-\infty}^{\infty} \frac{\sin(\sqrt\lambda\,x)\sin(\sqrt\lambda\,t)}{\lambda}\,d\sigma(\lambda).$$3

These are compatibility conditions on admissible spectral data: even after eigenvalues are fixed, the norming constants are not independent of the boundary parameters (Ashrafyan et al., 2017).

The derivation is a direct application of the Gelfand–Levitan equation for the transformation kernel $$F(x,t)=\int_{-\infty}^{\infty} \frac{\sin(\sqrt\lambda\,x)\sin(\sqrt\lambda\,t)}{\lambda}\,d\sigma(\lambda).$$4,

$$F(x,t)=\int_{-\infty}^{\infty} \frac{\sin(\sqrt\lambda\,x)\sin(\sqrt\lambda\,t)}{\lambda}\,d\sigma(\lambda).$$5

with diagonal value

$$F(x,t)=\int_{-\infty}^{\infty} \frac{\sin(\sqrt\lambda\,x)\sin(\sqrt\lambda\,t)}{\lambda}\,d\sigma(\lambda).$$6

In particular,

$$F(x,t)=\int_{-\infty}^{\infty} \frac{\sin(\sqrt\lambda\,x)\sin(\sqrt\lambda\,t)}{\lambda}\,d\sigma(\lambda).$$7

Since $$F(x,t)=\int_{-\infty}^{\infty} \frac{\sin(\sqrt\lambda\,x)\sin(\sqrt\lambda\,t)}{\lambda}\,d\sigma(\lambda).$$8 satisfies

$$F(x,t)=\int_{-\infty}^{\infty} \frac{\sin(\sqrt\lambda\,x)\sin(\sqrt\lambda\,t)}{\lambda}\,d\sigma(\lambda).$$9

evaluation at $\varphi(x,\lambda)$0 gives $\varphi(x,\lambda)$1, and $\varphi(x,\lambda)$2 is exactly the series appearing in the first sum rule. The second identity is obtained by reflection, replacing $\varphi(x,\lambda)$3 with $\varphi(x,\lambda)$4 and $\varphi(x,\lambda)$5 with $\varphi(x,\lambda)$6. In this setting, the Gelfand–Levitan condition is therefore a boundary-encoded spectral sum rule rather than an operator positivity statement (Ashrafyan et al., 2017).

5. Gel'fand–Levitan–Krein and Gelfand–Levitan–Marchenko operational conditions

In the GLK treatment of a one-dimensional coefficient inverse problem for

$\varphi(x,\lambda)$7

with

$\varphi(x,\lambda)$8

the paper again states that no separate named Gel'fand–Levitan condition is introduced. Instead, the method is formulated through a travel-time variable

$\varphi(x,\lambda)$9

an even extension of the transformed coefficient and solution, and a GLK integral equation

$$-\varphi''+q(x)\varphi=\lambda\varphi,\qquad \varphi(0,\lambda)=0,\qquad \varphi'(0,\lambda)=1,$$0

The paper isolates two conditions closest to a GLK admissibility criterion. First, the boundary normalization

$$-\varphi''+q(x)\varphi=\lambda\varphi,\qquad \varphi(0,\lambda)=0,\qquad \varphi'(0,\lambda)=1,$$1

implies the data compatibility relation

$$-\varphi''+q(x)\varphi=\lambda\varphi,\qquad \varphi(0,\lambda)=0,\qquad \varphi'(0,\lambda)=1,$$2

Second, Theorem 2 states that if $$-\varphi''+q(x)\varphi=\lambda\varphi,\qquad \varphi(0,\lambda)=0,\qquad \varphi'(0,\lambda)=1,$$3 is known on $$-\varphi''+q(x)\varphi=\lambda\varphi,\qquad \varphi(0,\lambda)=0,\qquad \varphi'(0,\lambda)=1,$$4 and the GLK equation has a unique solution for each $$-\varphi''+q(x)\varphi=\lambda\varphi,\qquad \varphi(0,\lambda)=0,\qquad \varphi'(0,\lambda)=1,$$5, then the coefficient inverse problem has a unique solution on $$-\varphi''+q(x)\varphi=\lambda\varphi,\qquad \varphi(0,\lambda)=0,\qquad \varphi'(0,\lambda)=1,$$6; conversely, if there exists a unique coefficient satisfying the assumptions and $$-\varphi''+q(x)\varphi=\lambda\varphi,\qquad \varphi(0,\lambda)=0,\qquad \varphi'(0,\lambda)=1,$$7, then the GLK equation has a unique solution. The discussion adds that existence and uniqueness are guaranteed only when the data belong to the range of the forward operator, that is, for errorless data (Karchevsky et al., 2013).

In Zakharov–Shabat inverse scattering, the left and right Gelfand–Levitan–Marchenko systems are used to reconstruct $$-\varphi''+q(x)\varphi=\lambda\varphi,\qquad \varphi(0,\lambda)=0,\qquad \varphi'(0,\lambda)=1,$$8 from scattering data. One source writes the left GLME as

$$-\varphi''+q(x)\varphi=\lambda\varphi,\qquad \varphi(0,\lambda)=0,\qquad \varphi'(0,\lambda)=1,$$9

$$\varphi(x,\lambda)=\frac{\sin(\sqrt\lambda\,x)}{\sqrt\lambda}+\int_0^x K(x,t)\frac{\sin(\sqrt\lambda\,t)}{\sqrt\lambda}\,dt,$$0

with

$$\varphi(x,\lambda)=\frac{\sin(\sqrt\lambda\,x)}{\sqrt\lambda}+\int_0^x K(x,t)\frac{\sin(\sqrt\lambda\,t)}{\sqrt\lambda}\,dt,$$1

and reconstruction formula

$$\varphi(x,\lambda)=\frac{\sin(\sqrt\lambda\,x)}{\sqrt\lambda}+\int_0^x K(x,t)\frac{\sin(\sqrt\lambda\,t)}{\sqrt\lambda}\,dt,$$2

After the substitution $$\varphi(x,\lambda)=\frac{\sin(\sqrt\lambda\,x)}{\sqrt\lambda}+\int_0^x K(x,t)\frac{\sin(\sqrt\lambda\,t)}{\sqrt\lambda}\,dt,$$3, the unknowns are defined on $$\varphi(x,\lambda)=\frac{\sin(\sqrt\lambda\,x)}{\sqrt\lambda}+\int_0^x K(x,t)\frac{\sin(\sqrt\lambda\,t)}{\sqrt\lambda}\,dt,$$4, and the infinite interval is truncated to $$\varphi(x,\lambda)=\frac{\sin(\sqrt\lambda\,x)}{\sqrt\lambda}+\int_0^x K(x,t)\frac{\sin(\sqrt\lambda\,t)}{\sqrt\lambda}\,dt,$$5. The paper explicitly says that it does not formulate a classical abstract GLM solvability theorem. Instead it introduces practical conditions: the kernel outside $$\varphi(x,\lambda)=\frac{\sin(\sqrt\lambda\,x)}{\sqrt\lambda}+\int_0^x K(x,t)\frac{\sin(\sqrt\lambda\,t)}{\sqrt\lambda}\,dt,$$6 must be negligible enough for the desired accuracy; exponentially growing discrete-spectrum contributions must be cut off; TIB-type recursions should start where the potential and relevant matrix elements are small; well-separated solitons require decomposition into local stability zones; and the empirically observed stability zone for a one-soliton signal extends roughly to distance about $$\varphi(x,\lambda)=\frac{\sin(\sqrt\lambda\,x)}{\sqrt\lambda}+\int_0^x K(x,t)\frac{\sin(\sqrt\lambda\,t)}{\sqrt\lambda}\,dt,$$7 from the soliton center, where $$\varphi(x,\lambda)=\frac{\sin(\sqrt\lambda\,x)}{\sqrt\lambda}+\int_0^x K(x,t)\frac{\sin(\sqrt\lambda\,t)}{\sqrt\lambda}\,dt,$$8 is the soliton amplitude (Medvedev et al., 2021).

A later high-order GLME paper makes the same terminological point: no standalone Gelfand–Levitan condition is defined. The explicit assumptions are that the potential $$\varphi(x,\lambda)=\frac{\sin(\sqrt\lambda\,x)}{\sqrt\lambda}+\int_0^x K(x,t)\frac{\sin(\sqrt\lambda\,t)}{\sqrt\lambda}\,dt,$$9 decays at least exponentially as $K(x,t)$0, that the scattering data are of the form

$K(x,t)$1

that the half-line integral is truncated to $K(x,t)$2 with $K(x,t)$3 sufficiently large, and that a uniform grid $K(x,t)$4 is used. The transformed operators are Hankel; after reversal they become Toeplitz; and the discretized system is solved through a block Levinson or inner-bordering scheme. The Gregory-weighted matrix has the structure

$K(x,t)$5

and the method relies on $K(x,t)$6. Here the practical analogue of a Gelfand–Levitan condition is therefore structural and numerical: admissible decaying data, finite-window truncation, and an almost-Toeplitz low-rank perturbation compatible with fast inversion via the Woodbury formula (Medvedev et al., 2024).

6. Adjacent terminology and common confusions

The phrase should be distinguished from several neighboring constructions. In the paper on a Gelfand–Levitan trace formula for finite quantum graphs, no standalone Gelfand–Levitan condition is defined. The relevant hypothesis is instead a genericity assumption on the self-adjoint coupling matrix,

$K(x,t)$7

which allows the vertex conditions to be rewritten in Hermitian form and excludes, among others, Dirichlet, standard/Kirchhoff, and $K(x,t)$8-coupling while including Robin, Neumann, and $K(x,t)$9-type couplings. This is a condition for a trace formula on quantum graphs, not the inverse-spectral admissibility condition of classical Gelfand–Levitan theory (Freitas et al., 2019).

It should also be distinguished from the Gelfand condition appearing in elliptic PDEs. In the two-parameter Gelfand-type system

$L^2(\mathbb R_+)$00

the paper studies regularity of extremal solutions and proves smoothness when $L^2(\mathbb R_+)$01 and

$L^2(\mathbb R_+)$02

That threshold quantifies closeness to the scalar diagonal case $L^2(\mathbb R_+)$03, but it is unrelated to the Gelfand–Levitan inverse-spectral condition (Cowan, 2010).

Finally, some applications use Gel'fand–Levitan–Marchenko theory without defining a corresponding condition at all. In the hypertriton calculation based on GLM-restored $L^2(\mathbb R_+)$04 and $L^2(\mathbb R_+)$05 potentials, the method takes theoretical sub-threshold scattering phase shifts as input and uses effective local central $L^2(\mathbb R_+)$06-wave information, but the paper explicitly states that the phrase “Gel'fand-Levitan condition” does not appear there (Meoto et al., 2019). This suggests that, outside core inverse-spectral theory, the phrase is often absent even when GL or GLM machinery is central.

Taken together, these sources support a precise encyclopedia-level conclusion. In classical inverse spectral theory, the Gelfand–Levitan condition is best understood as an admissibility criterion ensuring that inverse data generate a solvable integral equation and hence a potential. In Boundary Control form, that criterion becomes the positivity of the connecting operator $L^2(\mathbb R_+)$07. In regular Sturm–Liouville theory, it appears as exact sum rules tying norming constants to boundary parameters. In numerical GLK and GLME work, the same idea survives as a set of compatibility, truncation, and stability requirements rather than as a single named theorem (Avdonin et al., 29 May 2025).