The sharp cost of binding: Lieb–Thirring inequalities and their equality cases

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Family 262 · Mathematical physics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

An attractive potential can bind a quantum particle. If we give the potential a fixed budget, measured by an integral of a power of its strength, how much total binding can it create? Is one carefully shaped well more efficient than many interacting bound states?

The Lieb–Thirring inequality answers this with an upper bound on a sum of powers of negative energy levels. The outstanding issue in the one-dimensional range considered here is the best constant: an inequality can be valid while still losing some of the true efficiency.

Family 262 contains three claims. M3 gives the sharp scalar constant for the remaining exponents between one half and three halves. M2 extends that constant to every finite matrix size, allowing internal channels to mix as position changes. M1 classifies every equality case. Its conclusion is that exact optimality forces the channels to stop rotating: one constant basis separates independent scalar one-bound-state profiles.

2. Background and essential definitions

On the real line, consider the Schrödinger operator

$$H_W=-\frac{d^2}{dx^2}-W(x).$$

For a scalar potential, $W(x)\geq0$ describes attraction because it enters with a minus sign. For a matrix potential, a wave function takes values in $\mathbb C^m$, and $W(x)$ is an $m\times m$ Hermitian positive semidefinite matrix. Hermitian means $W^*=W$; positive semidefinite means $v^*Wv\geq0$ for every vector $v$.

The operator is defined through its energy form,

$$\int_{\mathbb R}\|\psi'(x)\|^2\,dx -\int_{\mathbb R}\langle\psi(x),W(x)\psi(x)\rangle\,dx.$$

Its natural form domain is $H^1$, the space of square-integrable wave functions with square-integrable weak derivative. This formulation accommodates measurable potentials rather than requiring a classical differential equation at every point.

Write the negative eigenvalues as $\lambda_j<0$, counted with multiplicity. Their magnitudes measure binding below the free-particle threshold zero. For $\gamma>0$, the spectral moment is

$$S_\gamma(W)=\sum_{\lambda_j<0}|\lambda_j|^\gamma.$$

The exponent changes how heavily deep states are weighted relative to shallow ones. The associated potential cost is

$$P_\gamma(W)=\int_{\mathbb R}\operatorname{tr} \bigl(W(x)^{\gamma+1/2}\bigr)\,dx.$$

Matrix powers are formed by applying the scalar power to each eigenvalue; the trace is the ordinary, unnormalized sum of diagonal entries. A sharp inequality asks for the smallest $C_\gamma$ such that $S_\gamma(W)\leq C_\gamma P_\gamma(W)$ for every admissible potential.

There are two natural benchmarks. A single optimally shaped bound state gives the one-state constant. A broad well with many levels gives a semiclassical constant, obtained by replacing spectral sums with phase-space integrals. The conjecture predicts which benchmark wins in each exponent range.

3. Development history and previous partial results

Lieb and Thirring introduced these inequalities in the 1970s in their work on fermionic kinetic energy and stability of matter. The one-state optimization has an earlier source in Keller’s 1961 work on Schrödinger eigenvalues. The full problem is harder because an arbitrary potential may have many, even infinitely many, negative levels.

At $\gamma=3/2$, inverse-scattering identities connected to the Korteweg–de Vries equation supplied the sharp scalar constant $3/16$. Aizenman and Lieb’s integration argument transfers sharp semiclassical estimates to larger exponents. Laptev and Weidl developed matrix and operator-valued versions and a dimension-lifting method. These are established foundations of the subject, not claims introduced by this family.

At the lower endpoint $\gamma=1/2$, Weidl established finiteness, and Hundertmark, Lieb and Thomas obtained the sharp scalar constant $1/2$. Hundertmark, Laptev and Weidl treated operator-valued extensions and improved bounds in the open interval. Work on orthonormal functions, including that of Dolbeault, Laptev and Loss, further narrowed the gap.

An especially important immediate predecessor is Read and Schulz’s September 2026 theorem. It establishes the sharp value

$$C_1=\frac{4}{3\sqrt3\,\pi}$$

at exponent one and includes an operator-valued extension. Both the scalar and matrix manuscripts explicitly acknowledge this priority. Their additional claimed exponent range is therefore $1/2<\gamma<3/2$ with $\gamma\ne1$, rather than a first solution of the exponent-one case.

Numerical studies had suggested concentration into a one-state profile or separation into distant copies. Such evidence does not control all possible orthonormal families. The claimed new action estimate addresses that uniform many-state obstruction, and the equality paper asks what exact saturation forces beyond the numerical value of the constant.

4. What the new manuscripts claim

The sharp scalar and matrix bounds

For every $1/2<\gamma<3/2$, every finite $m\geq1$, and every measurable positive semidefinite Hermitian $W$ with finite $P_\gamma(W)$, M2 claims

$$S_\gamma(W)\leq C_\gamma P_\gamma(W),$$

with

$$C_\gamma= \left(\frac{\gamma-1/2}{\gamma+1/2}\right)^{\gamma-1/2} \frac{\Gamma(\gamma+1)}{\sqrt\pi\,\Gamma(\gamma+3/2)}.$$

Here $\Gamma$ is the gamma function, the extension of factorial satisfying $\Gamma(n+1)=n!$ for nonnegative integers. The formula is the scalar one-state constant, independent of matrix size.

All negative eigenvalues are included. The potential can be rough and unbounded, and its matrices need not commute at different positions. In particular, the theorem is not restricted to a potential that is already diagonal in one fixed basis. M3 gives the scalar case, including the full negative spectral sum, and a separate signed-potential extension on its natural form domain. The displayed matrix theorem uses positive semidefinite potentials.

Every equality case

Set

$$r=(\gamma-1/2)^{-1}>1.$$

M1 claims that equality holds precisely when, almost everywhere,

$$W(x)=U\operatorname{diag}(w_1(x),\ldots,w_k(x),0,\ldots,0)U^*,$$

where $0\leq k\leq m$, $U$ is a constant unitary matrix, and

$$w_j(x)=(r+1)a_j^2\operatorname{sech}^2\bigl(ra_j(x-x_j)\bigr).$$

The scale $a_j$ is positive, the center $x_j$ is real, and $\operatorname{sech}t=1/\cosh t$. Each nonzero channel has exactly one negative eigenvalue, $-a_j^2$. The parameters can be chosen independently; repeated scales and coincident profiles are allowed. The case $k=0$ includes the zero potential.

Thus an extremal matrix potential has at most $m$ negative eigenvalues. This is a conclusion, not an assumption used to exclude infinitely many levels at the start. In the scalar case it leaves precisely one nonzero bound state, up to translation and scaling.

The endpoint $\gamma=3/2$ is excluded from this classification. There the equality structure is different: scalar multisolitons can support several bound states in a single channel. Extending the interior formula to the endpoint would lose that distinction.

5. Why the advance matters

Provisional importance: 5/5. Sharp Lieb–Thirring constants are central quantities in spectral theory and mathematical physics. The claimed resolution completes the remaining one-dimensional open-interval cases and simultaneously shows that noncommuting finite internal channels cannot improve the optimal constant.

The equality theorem supplies more than an attractive example. It says that exact optimality removes a whole class of possible behavior: position-dependent channel rotation and multiple bound states in one scalar channel. This turns an optimization problem into a structural classification.

One can also see why the matrix result is substantial. Diagonalizing $W(x)$ separately at each point does not reduce the problem to independent scalar operators. A changing basis contributes derivative terms to the kinetic energy. The theorem must control that extra freedom rather than silently discard it. Its dimension-independent constant gives the same cost-to-binding limit at every finite channel count.

6. The main idea

The source papers begin on a bounded interval with finitely many orthonormal trial functions. Their individual binding scales are recorded in a diagonal matrix $K$, whose entries are square roots of positive binding energies. A lower triangular coefficient matrix combines the trial functions while preserving their successive spans. This matters because spectral inequalities remain valid on those nested spans.

The central object is an action: a functional containing kinetic energy, the chosen spectral scales, and a nonlinear density penalty. The papers bound this action from two sides. The eigenfunction equations give an upper comparison involving the potential cost. A specially constructed matrix path gives a lower comparison involving the desired sum of eigenvalue moments.

The path joins $K$ to $-K$. A regularized matrix field and a continuation argument arrange the necessary endpoint conditions. An integrated resolvent comparison controls the nonlinear term. In the scalar-potential paper, the relevant pointwise density has rank at most one. The matrix paper’s additional estimate allows arbitrary rank and retains the noncommutative terms that the scalar argument does not encounter.

Passing from finite intervals to the line, and from finite lists to the full spectral sum, is part of the claim. It is why finding the explicit optimizing well alone would be insufficient: an arbitrary potential might bind many shallow states whose collective contribution matters.

For equality, M1 retains nonnegative errors instead of throwing them away after obtaining an inequality. Zero total deficit forces these errors to vanish. Some measure failure of first-order differential relations; others measure failure of matrix commutation and density identities.

A key compactness observation is that an increasing list of trial states still has only $m$ internal columns. The retained errors control the contribution of shallow tail states, allowing the equality analysis to begin without assuming a finite spectrum. The resulting identities separate different energy blocks, force their row spaces to be independent of position, and leave at most $m$ orthogonal directions.

Within each fixed channel, a Riccati-type first-order equation determines the hyperbolic-secant profile. The explicit formula in section 4 is therefore the endpoint of the claimed rigidity argument, rather than an ansatz imposed on all competitors.

7. Scope, limitations, and open questions

The new matrix theorem and complete equality classification are stated for finite matrix size and the strict exponent interval. A constant independent of $m$ does not by itself constitute the paper’s proof of every infinite-dimensional operator-valued extension. Earlier operator-valued results at particular exponents retain their own hypotheses.

This family does not settle sharp Lieb–Thirring constants in every spatial dimension. The relation between one-state and semiclassical optimization changes with dimension and exponent. Likewise, the scalar signed-potential statement should not be substituted for the positive semidefinite assumption in the matrix theorem without consulting its separate formulation.

Exact equality also differs from quantitative stability. The classification does not itself provide a universal estimate saying how close a nearly optimal potential must be to a direct sum of solitons. Widely separated profiles and independent scales suggest why such a question requires careful choice of distance and normalization.

8. Sources and further reading

Read–Schulz’s primary paper is linked in section 3. The sources’ bibliographies identify Keller’s Lower bounds and isoperimetric inequalities for eigenvalues of the Schrödinger equation (1961), Lieb–Thirring’s original spectral and kinetic-energy papers, Hundertmark–Lieb–Thomas’s sharp lower-endpoint result, and Laptev–Weidl’s Sharp Lieb–Thirring inequalities in high dimensions (2000). M3 explains the scalar action method; M2 isolates its arbitrary-rank extension; M1 retains the equality defects and derives the fixed-channel classification.

Source manuscripts

• M1. OpenAI, Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality (October 5, 2026).
• M2. OpenAI, Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials (October 5, 2026).
• M3. OpenAI, Sharp one-dimensional Lieb–Thirring constants (September 23, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.