Let V and W be real or complex finite dimensional vector spaces with inner products 〈·, ·〉V and 〈·, ·〉W , respectively. Let L : V → W be linear. If there is a transformation L∗ : W → V for which 〈Lv,w〉W = 〈v, Lw〉V (1) holds for every pair of vectors v ∈ V and w in W , then L∗ is said to be the adjoint of L. Some of the properties of L∗ are listed below. Proposition 1.1. Let L : V →W be linear. ...

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We establish sufficiency conditions in order to achieve approximation to discrete eigenvalues of self-adjoint operators in the second-order projection method suggested recently by Levitin and Shargorodsky, [15]. We find explicit estimates for the eigenvalue error and study in detail two concrete model examples. Our results show that, unlike the majority of the standard methods, second-order pro...

We construct efficient approximations for the eigenfunctions of non-selfadjoint Schrödinger operators in one dimension. The same ideas also apply to the study of resonances of self-adjoint Schrödinger operators which have dilation analytic potentials. In spite of the fact that such eigenfunctions can have surprisingly complicated structures with multiple local maxima, we show that a suitable ad...