The one dimensional Schrödinger hydrogen atom is an interesting mathematical and physical problem to study bound states, eigenfunctions and quantum degeneracy issues. This 1D physical system gave rise to some intriguing controversy over more than four decades. Presently, still no definite consensus seems to have been reached. We reanalyzed this apparently controversial problem, approaching it from a Fourier transform representation method combined with some fundamental (basic) ideas found in self-adjoint extensions of symmetric operators. In disagreement with some previous claims, we found that the complete Balmer energy spectrum is obtained together with an odd parity set of eigenfunctions. Closed form solutions in both coordinate and momentum spaces were obtained. No twofold degeneracy was observed as predicted by the degeneracy theorem in one dimension, though it does not necessarily have to hold for potentials with singularities. No ground state with infinite energy exists since the corresponding eigenfunction does not satisfy the Schrödinger equation at the origin. PACS number(s): 03.65.Ge 03.65.-w *

Generalized universality, as recently proposed, postulates a universal non-Gaussian form of the probability density function (PDF) of certain global observables for a wide class of highly correlated systems of finite volume N . Studying the 2D XY -model, we link its validity to renormalization group properties. It would be valid if there were a single dimension 0 operator, but the actual existence of several such operators leads to T -dependent corrections. The PDF is the Fourier transform of the partition function Z(q) of an auxiliary theory which differs by a dimension 0 perturbation with a very small imaginary coefficient iq/N from a theory which is asymptotically free in the infrared. We compute the PDF from a systematic loop expansion of ln Z(q).PACS numbers: 05.70. Jk, 05.50.+q, 75.10.Hk, 68.35.Rh Trying to understand universality properties of critical systems near a second order phase transition was extremely fruitful for the development of physics. It lead to the development of Wilsons renormalization group (RG) [1] which found applications ranging from solid state physics to elementary particle theory. Universality classes in the RG-sense depend on the dimension of the system and on the symmetry properties of the order parameter.Recently a generalized universality has been proposed [2], which goes far beyond the known picture. It is therefore of great general interest to understand the conditions for its validity. Generalized universality is supposed to hold true for systems sharing the properties of strong correlations, finite volume and self similarity, no matter whether they have the same symmetries and dimensions nor whether they correspond to equilibrium or non-equilibrium systems. Such new universality is expressed in a non-Gaussian universal curve these systems share when the PDF of some global quantity of each system is plotted (universal fluctuations). Studying the 2D XY-model as an example, we link its validity to RGproperties of the system, viz. the existence of dimension 0 perturbations of a system which is asymptotically free in the infrared. It would be exactly true if there were one single dimension 0 operator. But the actual existence of several dimension 0 operators in the model leads to T-dependent corrections to the supposedly universal curve which do not go away when the volume N → ∞. * Electronic address: gerhard.mack@desy.de † Electronic address: gpalma@lauca.usach.cl ‡ Electronic address: lvergara@lauca.usach.cl, If there were no dimension 0 operator, the PDF would be Gaussian.Let us recall some previous results. The supposed universal curve for the PDF, which we call BHP, is nonGaussian in spite of the result that a naive application of the central limit theorem would have given; this possibility has been attributed [2] to the anomalous statistical property of finite size critical systems that, whatever their size is, cannot be divided into mesoscopic regions that are statistically independent, a necessary condition for the central limit theorem to apply. In Ref. [3] it was s...

Using an improved estimator in the loop-cluster algorithm, we investigate the constraint effective potential of the magnetization in the spin 1 2 quantum XY model. The numerical results are in excellent agreement with the predictions of the corresponding low-energy effective field theory. After its low-energy parameters have been determined with better than permille precision, the effective theory makes accurate predictions for the constraint effective potential which are in excellent agreement with the Monte Carlo data. This shows that the effective theory indeed describes the physics in the low-energy regime quantitatively correctly.

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In recent works [S. T. Bramwell, P. C. W. Holdsworth, and J.-F. Pinton, Nature (London) 396, 552 (1998); S. T. Bramwell et al., Phys. Rev. Lett. 84, 3744 (2000)], a generalized universality has been proposed, linking phenomena as dissimilar as two-dimensional (2D) magnetism and turbulence. To test these ideas, we performed Monte Carlo simulations of the 2D XY model. We found that the shape of the probability distribution function for the magnetization M is non-Gaussian and independent of the system size-in the range of the lattice sizes studied-below the Kosterlitz-Thoules temperature. However, our results suggest that in the full 2D XY model the shape of these distributions has a slight dependence on temperature-for finite volume-below the lattice-shifted critical temperature T*(L). This behavior can be explained by using renormalization group arguments and an extended finite-size scaling analysis, and by the existence of bounds for M.