ar X iv : m at h - ph / 0 51 10 29 v 1 8 N ov 2 00 5 CONVERGENCE OF SCHRÖDINGER OPERATORS
نویسندگان
چکیده
For a large class, containing the Kato class, of real-valued Radon measures m on R d the operators −∆ + ε 2 ∆ 2 + m in L 2 (R d , dx) tend to the operator −∆ + m in the norm resolvent sense, as ε tends to zero. If d ≤ 3 and a sequence (µ n) of finite real-valued Radon measures on R d converges to the finite real-valued Radon measure m weakly and, in addition, sup n∈N µ ± n (R d) < ∞, then the operators −∆ + ε 2 ∆ 2 + µ n converge to −∆ + ε 2 ∆ 2 + m in the norm resolvent sense. Explicit upper bounds for the rates of convergences are derived. One can choose point measures µ n with mass at only finitely many points so that a combination of both convergence results leads to an efficient method for the numerical computation of the eigenvalues in the discrete spectrum and corresponding eigenfunctions of Schrödinger operators. Weak convergence of potentials implies norm-resolvent convergence of the corresponding one-dimensional Schrödinger operators. This result from [5] may be interesting for several reasons. For instance every finite real-valued Radon measure on R is the weak limit of a sequence of point measures with mass at only finitely many points. There exist efficient numerical methods for the computation of the eigenvalues and corresponding eigenfunctions of one-dimensional Schrödinger operators with a potential supported by a finite set; actually the effort for the computation of an eigenvalue 1
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We prove two limit relations between Schrödinger operators perturbed by measures. First, weak convergence of finite real-valued Radon measures µ n −→ m implies that the operators −∆ + ε 2 ∆ 2 + µ n in L 2 (R d , dx) converge to −∆ + ε 2 ∆ 2 + m in the norm resolvent sense, provided d ≤ 3. Second, for a large family, including the Kato class, of real-valued Radon measures m, the operators −∆ + ε...
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