# A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator

@article{Hundertmark1998ASB, title={A sharp bound for an eigenvalue moment of the one-dimensional Schr{\"o}dinger operator}, author={Dirk Hundertmark and Elliott H. Lieb and Lawrence E. Thomas}, journal={Advances in Theoretical and Mathematical Physics}, year={1998}, volume={2}, pages={329-341} }

We give a proof of the Lieb-Thirring inequality in the critical case d=1, γ = 1/2, which yields the best possible constant.

## 58 Citations

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This paper has been withdrawn by the author in favor of a stronger result proven by the author with R. Frank and T. Weidl in arXiv:0707.0998

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We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schrödinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To…

## References

SHOWING 1-10 OF 22 REFERENCES

### The bound state of weakly coupled Schrödinger operators in one and two dimensions

- Mathematics, Computer Science
- 1976

### On the Lieb-Thirring constants L ?,1 for ??1/2

- Mathematics
- 1996

Let E i (H) denote the negative eigenvalues of the one-for the "limit" case = 1=2: This will imply improved estimates for the best constants L ;1 in (1) as 1=2 < < 3=2: 0. Let H = ??V denote the…

### Bounds on the eigenvalues of the Laplace and Schroedinger operators

- Mathematics
- 1976

If 12 is an open set in R", and if N(£l, X) is the number of eigenvalues of A (with Dirichlet boundary conditions on d£2) which are < X (k > 0), one has the asymptotic formula of Weyl [1] , [2] : l i…

### On the Lieb-Thirring constantsLγ,1 for γ≧1/2

- Mathematics
- 1995

AbstractLetEi(H) denote the negative eigenvalues of the one-dimensional Schrödinger operatorHu≔−u″−Vu,V≧0, onL2(∝). We prove the inequality(1)
$$\mathop \sum \limits_i |E_i (H)|^{ \gamma } \leqq…

### Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter

- Mathematics
- 1975

We first prove that Σ |e (V)|, the sum of the negative energies of a single particle in a potential V, is bounded above by (4/15 π )∫|V|5/2. This in turn, implies a lower bound for the kinetic energy…

### Quantum mechanics: Non-relativistic theory,

- Physics
- 1958

The basic concepts of quantum mechanics Energy and momentum Schrodinger's equation Angular momentum Perturbation theory Spin The identity of particles The atom The theory of symmetry Polyatomic…

### The Number of Bound States of One-Body Schroedinger Operators and the Weyl Problem (代数解析学の最近の発展)

- Mathematics
- 1979

If N ((Ω,λ) is the number of eigenvalues of -Δ in a domain Ω, in a suitable Riemannian manifold of dimension n, we derive bounds of the form \(\tilde N(\Omega ,\lambda ) \le {D_n}{\lambda…

### Quantum mechanics of atoms and molecules

- Physics
- 1981

The third volume of this textbook on mathematical physics is devoted to quantum mechanics and especially to its applications to atomic systems. The author builds the theory on an axiomatic basis and…

### A new proof of the Cwikel-Lieb-Rosenblijum bound

- Mathematics
- 1985

which C is a constant and V_ denotes the negative part of V. The inequality (1.1) was derived in three quite different ways by Lieb [5], Cwikel [1] and Rosenbljum [8]. The best value for the constant…