Bargmann-Type Inequality for Half-Linear Differential Operators
© G. Bognár and O. Došlý. 2009
Received: 7 May 2009
Accepted: 21 August 2009
Published: 28 September 2009
The classical Bargmann inequality  originates from the nonrelativistic quantum mechanics and gives an upper bound for the number of bound states produced by a radially symmetric potential in the two-body system. In the subsequent papers, various proofs and reformulations of this inequality have been presented, we refer to [2, Chapter XIII], and to [3–5] for some details.
with the subcritical coefficient is disconjugate in , that is, any nontrivial solution of (1.3) has at most one zero in this interval. Hence, if the equation , with given by (1.1), has a solution with at least positive zeros, the perturbation function must be "sufficiently positive" in view of the Sturmian comparison theorem. Inequality (1.2) specifies exactly what "sufficient positiveness" means.
In physical sciences, there are known phenomena which can be described by differential equations with the so-called -Laplacian , see, for example, . If the potential in such an equation is radially symmetric, this equation can be reduced to a half-linear equation of the form (1.4).
where is a continuous function, and shows that if is the so-called subcritical coefficient, that is, , and there exists a solution of (1.6) with at least zeros in , then the integral satisfies an inequality which reduces to (1.2) in the linear case .
In this short section, we present some elements of the half-linear oscillation theory which we need in the proof of our main result. As we have mentioned in the previous section, the linear and half-linear oscillation theories are in many aspects very similar, so (1.4) can be classified as oscillatory or nonoscillatory as in the linear case.
If (1.4) is nonoscillatory, that is, (2.1) possesses a solution which exists on some interval , among all such solutions of (2.1), there exists the minimal one , minimal in the sense that any other solution of (2.1) which exists on some interval satisfies in this interval, see [9, 10] for details.
and is the minimal solution of this equation. A detailed study of half-linear Euler equation and of its perturbations can be found in .
3. Bargmann's Type Inequality
In this section, we present our main results, the half-linear version of Bargmann's inequality. We are motivated by the work in  where a short proof of this inequality based on the Riccati technique is presented. Here we show that this method, properly modified, can also be applied to (1.6).
which we needed to prove.
The authors thank the referees for their valuable remarks and suggestions which contributed substantially to the present version of the paper. The first author is supported by the Grant OTKA CK80228 and the second author is supported by the Research Project MSM0021622409 of the Ministry of Education of the Czech Republic and the Grant 201/08/0469 of the Grant Agency of the Czech Republic.
- Bargmann V: On the number of bound states in a central field of force. Proceedings of the National Academy of Sciences of the United States of America 1952, 38: 961–966. 10.1073/pnas.38.11.961MathSciNetView ArticleMATHGoogle Scholar
- Reed M, Simon B: Methods of Modern Mathematical Physics, Vol. IV. Analysis of Operators. Academic Press, Boston, Mass, USA; 1978.MATHGoogle Scholar
- Blanchard Ph, Stubbe J: Bound states for Schrödinger Hamiltonians: phase space methods and applications. Reviews in Mathematical Physics 1996,8(4):503–547. 10.1142/S0129055X96000172MathSciNetView ArticleMATHGoogle Scholar
- Schmidt KM: A short proof for Bargmann-type inequalities. The Royal Society of London 2002,458(2027):2829–2832. 10.1098/rspa.2002.1021MathSciNetView ArticleMATHGoogle Scholar
- Setô N: Bargmann's inequalities in spaces of arbitrary dimension. Publications of the Research Institute for Mathematical Sciences. Kyoto University 1974, 9: 429–461.View ArticleMATHGoogle Scholar
- Díaz JI: Nonlinear Partial Differential Equations and Free Boundaries. Vol. I: Elliptic Equations, Research Notes in Mathematics. Volume 106. Pitman, Boston, Mass, USA; 1985:vii+323.Google Scholar
- Agarwal RP, Grace SR, O'Regan D: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2002:xiv+672.View ArticleMATHGoogle Scholar
- Došlý O, Řehák P: Half-Linear Differential Equations, North-Holland Mathematics Studies. Volume 202. Elsevier, Amsterdam, The Netherlands; 2005:xiv+517.MATHGoogle Scholar
- Elbert Á, Kusano T: Principal solutions of non-oscillatory half-linear differential equations. Advances in Mathematical Sciences and Applications 1998, 18: 745–759.MathSciNetMATHGoogle Scholar
- Mirzov JD: Principal and nonprincipal solutions of a nonlinear system. Tbilisskiĭ Gosudarstvennyĭ Universitet. Institut Prikladnoĭ Matematiki. Trudy 1988, 31: 100–117.MathSciNetMATHGoogle Scholar
- Elbert Á, Schneider A: Perturbations of the half-linear Euler differential equation. Results in Mathematics 2000,37(1–2):56–83.MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.