On the point spectrum of Schrödinger operators

On the Point Spectrum of Positive Operators

1. Recently, G.-C. Rota proved the following result: Let (S, 2, p) be a measure space of finite measure, P a positive linear operator on Lx(S, 2, u) with Li-norm and L„-norm at most one. If a, | a\ = 1, is an eigenvalue of P such that af=Pf (JELx), then a2 is an eigenvalue such that a2|/|g"=P(|/|g"), where/=|/|g. It can be added that an|/|gn = P(|/|gn) for every integer n; thus Rota proved for ...

Locating the Spectrum for Magnetic Dirac and Schrödinger Operators

Spectrum and essential spectrum of linear combinations of composition operators on the Hardy space H2

Let -----. For an analytic self-map ---  of --- , Let --- be the composition operator with composite map ---  so that ----. Let ---  be a bounded analytic function on --- . The weighted composition operator ---  is defined by --- . Suppose that ---  is the Hardy space, consisting of all analytic functions defined on --- , whose Maclaurin cofficients are square summable. .....

Convergence of Schrödinger Operators

For a large class, containing the Kato class, of real-valued Radon measures m on R the operators −∆ + ε∆ + m in L(R, dx) tend to the operator −∆ +m in the norm resolvent sense, as ε tends to zero. If d ≤ 3 and a sequence (μn) of finite real-valued Radon measures on R converges to the finite real-valued Radon measure m weakly and, in addition, supn∈N μ ± n (R) < ∞, then the operators −∆ + ε∆ + μ...

Some Schrödinger Operators with Dense Point Spectrum

Given any sequence {En}n−1 of positive energies and any monotone function g(r) on (0,∞) with g(0) = 1, lim r→∞ g(r) = ∞, we can find a potential V (x) on (−∞,∞) so that {En}n=1 are eigenvalues of − d 2 dx2 + V (x) and |V (x)| ≤ (|x| + 1)−1g(|x|). In [7], Naboko proved the following: Theorem 1. Let {κn}∞n=1 be a sequence of rationally independent positive reals. Let g(r) be a monotone function o...