# Calogero Type Bounds in Two Dimensions

@article{Laptev2021CalogeroTB, title={Calogero Type Bounds in Two Dimensions}, author={Ari Laptev and Larry Read and Lukas Schimmer}, journal={Archive for Rational Mechanics and Analysis}, year={2021}, volume={245}, pages={1491 - 1505} }

For a Schrödinger operator on the plane R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document} with electric potential V and an Aharonov–Bohm magnetic field, we obtain an upper bound on the number of its negative eigenvalues in terms of the L1(R2)\documentclass[12pt]{minimal} \usepackage…

## One Citation

### Weighted CLR type bounds in two dimensions

- Mathematics
- 2023

We derive weighted versions of the Cwikel-Lieb-Rozenblum inequality for the Schr\"odinger operator in two dimensions with a nontrivial Aharonov-Bohm magnetic field. Our bounds capture the optimal…

## References

SHOWING 1-10 OF 29 REFERENCES

### Inequalities for the Moments of the Eigenvalues of the Schrodinger Hamiltonian and Their Relation to Sobolev Inequalities

- Mathematics
- 2002

Estimates for the number of bound states and their energies, ej ≤ 0, are of obvious importance for the investigation of quantum mechanical Hamiltonians. If the latter are of the single particle form…

### On spectral estimates for two-dimensional Schrodinger operators

- Mathematics
- 2012

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V,\ V\ge 0,$ we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues (bound states), as the…

### Sharp Lieb-Thirring inequalities in high dimensions

- Mathematics
- 1999

We show how a matrix version of the Buslaev-Faddeev-Zakharov trace formulae for a one-dimensional Schr\"odinger operator leads to Lieb-Thirring inequalities with sharp constants $L^{cl}_{\gamma,d}$…

### Bargmann type estimates of the counting function for general Schrödinger operators

- Mathematics
- 2012

The paper concerns upper and lower estimates for the number of negative eigenvalues of one- and two-dimensional Schrödinger operators and more general operators with the spectral dimensions d ⩽ 2.…

### 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…

### An estimate for the number of bound states of the Schrodinger operator in two dimensions

- Mathematics
- 2003

For the Schrodinger operator -Δ + V on R^2 be the number of bound states. One obtains the following estimate: N(V) ≤ 1 + ∫_(R^2)∫_(R^2)|V(x)|V(y)|C_(1)ln|x-y|+C_2|^2 dx dy where C_1 = -1/2π and C_2 =…

### 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…

### Negative Eigenvalues of Two-Dimensional Schrödinger Operators

- Mathematics
- 2011

We prove a certain upper bound for the number of negative eigenvalues of the Schrödinger operator H = −Δ − V in $${\mathbb{R}^{2}.}$$R2.

### On the number of negative eigenvalues of Schrödinger operators with an Aharonov–Bohm magnetic field

- MathematicsProceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
- 2001

It is proved that for V+=max(V,0) in the subspace L1(R+ ; L∞(S1); r dr) of L1(R2), there is a Cwikel–Lieb–Rosenblum–type inequality for the number of negative eigen2 R values of the operator ((1/i)∇…

### Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities

- Mathematics
- 2022

The analysis of eigenvalues of Laplace and Schrödinger operators is an important and classical topic in mathematical physics with many applications. This book presents a thorough introduction to the…