Abstract. We study the SVD of an arbitrary matrix , especially its subspaces of activation, which leads in natural manner to pseudoinverse of Moore-Bjenhammar-Penrose. Besides, we analyze the compatibility of linear systems and the uniqueness of the corresponding solution, and our approach gives the Lanczos classification for these systems.
We employ the local and isometric embedding of Riemannian spaces to construct one Lanczos potential in Gödel geometry.
No abstract
We show that the matrix elements m|e βx |n for the one-dimensional harmonic oscillator permit to resolve the vibrational Schrödinger equation for the Morse interaction. Keywords:Morse potential, one-dimensional harmonic oscillator, matrix elements PACS: 02.10.Yn, 03.65.Ge, 03.65.Fd IntroductionIn [1][2][3] were calculated the matrix elementsfor the harmonic oscillator (HO) in one dimension, where β ≥ 0 is an arbitrary parameter. Thus, it was obtained the following result for m ≥ n:in terms of the associated Laguerre polynomials L q n . It is interesting to observe that the 2th order differential equation defining to L q n (see [4] p. 781) permits to prove via (2) that f (β) satisfies the equationwhere A = m + n + 1 and Q = (m − n) 2 ; that is, (2) is a solution of (3). In Sec. 2, f (β) is employed to resolve the radial Schrödinger equation for the Morse potential. Radial wave function for the Morse potentialMorse [5][6][7] proposed the potentialas an approximation to vibrational motion of a diatomic molecule, where D is the dissociation energy (well depth), r 0 is the nuclear equilibrium separation, and a is a parameter associated with the well width, such that a √ 2D/(2π)gives the frequency of small classical vibrations around r 0 . If we make the change of variable u = r − r 0 and we use natural units, then the corresponding Schrödinger equation iswhere ψ M /r is the Morse's radial wave function. If now at (5) we introduce a new independent variable β given bythen (5) adopts the formwith the same structure as (3)! Therefore by formal comparison of (3) with (7) we have:which implies that m = n is not possible because in this case the value E = 0 is forbidden for bound Caltenco et al. / Lithuanian J. Phys. 50, 403-404 (2010) states; then from (8) results K > 1 which is the condition [5] for the existence of a discrete spectrum energy. Besides, as E n = 0 and K > 1, then (8) leads tothis means [8] a finite number of bound states. From (3) and (7) is clear that ψ M is proportional to f (β) given by (2) then:where q = K e −a(r−r 0 ) and b = m − n = K − 2n − 1, in accordance with [9] for ψ M /r normalized to unity. Thus, we see that the Schrödinger equation was easily resolved, for the vibrational Morse oscillator, using the matrix elements m|e βx |n for the one-dimensional HO. This is a one more sample of the multiple correspondences [7,10,11] between the Morse and harmonic oscillators. Our results (8) and (10)
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