zbMATH — the first resource for mathematics

On the localization of binding for Schrödinger operators and its extensions to elliptic operators. (English) Zbl 0859.35082
The lowest eigenvalue \(E(R)\) of perturbations of the \(n\)-dimensional Laplacian \(-\Delta+V(x)+W(x-R)\) is studied for large \(R\). The quantity is important in the discussion of the Efimov effect [H. Tamura, J. Funct. Anal. 95, No.. 2, 433-459 (1991; Zbl 0761.35078)]. The main results are the bounds \(-C_1\leq R^{n-2}E(R)\leq-C_2\) in the case \(n\geq 5\) with assumptions on the decay of \(V\) and \(W\) which are in some sense optimal. The results are generalized to perturbations of equivariant operators for more general group actions on a manifold.
Reviewer: J.Asch (Marseille)

35P15 Estimates of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35B20 Perturbations in context of PDEs
35Q40 PDEs in connection with quantum mechanics
Full Text: DOI
[1] S. Agmon,Bounds on exponential decay of eigenfunctions of Schrödinger operators, inSchrödinger Operators, ed. S. Graffi, Lecture Notes in Math., Vol. 1159, Springer-Verlag, Berlin, 1985, pp. 1–38. · Zbl 0583.35027
[2] W. D. Evans, R. T. Lewis and Y. Saitō,Zhislin’s theorem revisited, J. Analyse Math.58 (1992), 191–212. · Zbl 0808.35097 · doi:10.1007/BF02790364
[3] M. Klaus and B. Simon,Binding of Schrödinger particles through conspiracy of potential wells, Ann. Inst. Henri Poincaré, Sect. A: Physique théorique30 (1979), 83–87.
[4] V. Lin and Y. Pinchover,Manifolds with group actions and elliptic operators, Memoirs Amer. Math. Soc., Vol. 112, No. 540 (1994). · Zbl 0816.58041
[5] R. D. Nussbaum and Y. Pinchover,On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications, J. Analyse Math.59 (1992), 161–177. · Zbl 0816.35095 · doi:10.1007/BF02790223
[6] Yu. N. Ovchinnikov and I. M. Sigal,Number of bound states of three-body systems and Efimov’s effect, Ann. Phys.123 (1979), 274–295. · doi:10.1016/0003-4916(79)90339-7
[7] Y. Pinchover,Criticality and ground states for second-order elliptic equations, J. Differential Equations80 (1989), 237–250. · Zbl 0697.35036 · doi:10.1016/0022-0396(89)90083-1
[8] Y. Pinchover,On criticality and ground states for second-order elliptic equations, II, J. Differential Equations87 (1990), 353–364. · Zbl 0714.35055 · doi:10.1016/0022-0396(90)90007-C
[9] Y. Pinchover,On the equivalence of Green functions of second order elliptic equations in \(\mathbb{R}\) n, Differential and Integral Equations5 (1992), 481–490. · Zbl 0772.35015
[10] B. Simon,Brownian motion, LP properties of Schrödinger operators and the localization of binding, J. Funct. Anal.35 (1980), 215–229. · Zbl 0446.47041 · doi:10.1016/0022-1236(80)90006-3
[11] H. Tamura,Existence of bound states for double well potentials and the Efimov effect, inFunctional-Analytic Methods for Partial Differential Equations, eds. H. Fujita, T. Ikebe and S. T. Kuroda, Lecture Notes in Math., Vol. 1450, Springer-Verlag, Berlin, 1990, pp. 173–186. · Zbl 0757.47034
[12] H. Tamura,The Efimov effect of three-body Schrödinger operators, J. Funct. Anal.95 (1991), 433–459. · Zbl 0761.35078 · doi:10.1016/0022-1236(91)90038-7
[13] H. Tamura,The Efimov effect of three-body Schrödinger operators: Asymptotics for the number of negative eigenvalues, Nagoya Math. J.130 (1993), 55–83. · Zbl 0827.35099
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.