INTRODUCTION: In the past five years, researchers have increasingly turned to the study of mental health outcomes in LGBt populations. the present paper summarizes recent literature on the relationship between minority stress experienced by sexual minorities and mental health. eViDeNce acQUisitiON: PsyciNFO, PubMed, and the eBscO Psychology and Behavioral science collection were searched for papers concerning minority stress and mental health disparities in LGBt populations, published between 1 January 2014 and 30 June 2018. all collected papers were screened using the following criteria: study involving >50 individuals; written in english; focusing on clinical outcomes of depression, suicidality, and substance use in relation to experienced minority stress. eViDeNce sYNthesis: sixty-two papers were included in this review. Findings are reported under three main headings: studies primarily focused on depression, studies concerning suicidality and suicide attempts, and papers analyzing the correlation between substance use and minority stress in LGBt populations. the included studies supported the minority stress model as a framework to better explain disparities in mental health outcomes in sexual minority populations. higher rates of depression, suicidality, and substance use are reported in LGBt populations, as are the related minority stressors analyzed. cONcLUsiONs: sexual minorities still face numerous mental health disparities. research indicates that the levels of minority stressors positively predict mental health outcomes. Specific policies designed to support the civil rights of sexual minorities may help to overcome such inequalities.

Abstract. Here we consider stationary states for nonlinear Schrödinger equations with symmetric double well potentials. These stationary states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a simple pitch-fork bifurcation, and a couple of saddle points which unstable branches collapse in an inverse pitch-fork bifurcation. In this paper we show that in the semiclassical limit, or when the barrier between the two wells is large enough, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value (2); in contrast, when the nonlinearity power is larger than such a critical value then we always observe the second scenario. The remarkable fact is that such a critical value is an universal constant in the sense that it does not depends on the shape of the double well potential.Spontaneous symmetry breaking phenomenon is a rather important effect that arises in a wide range of physical systems modeled by nonlinear equations. In classical physics spontaneous symmetry breaking occurs in optics, and it has been experimentally observed for laser beams in Kerr media and focusing nonlinearity [1,2]. Another natural setting in which spontaneous symmetry breaking phenomenon may arise is for Bose Einstein condensates with an effective double well formed by the combined effect of a parabolic-like trap and a periodical-like optical lattice [3,4,5]. Also, the study of gases of pyramidal molecules, like the ammonia N H 3 , it is a topic where spontaneous symmetry breaking phenomenon actually plays a crucial role. In [6,7] has been introduced a nonlinear mean field model of a gas of pyramidal molecules; in this model spontaneous symmetry breaking explaining the presence of two asymmetrical degenerate ground states, corresponding to the different localization of the molecules, has been predicted with a full agreement with experimental data [7,8].The n-dimensional linear Schrödinger equation with a symmetric potential with double well shape has stationary states of a definite even and odd-parity. However, the introduction of a nonlinear term (which usually models, in quantum mechanics, an interacting many-particle system) may give rise to asymmetrical states related to a spontaneous symmetry breaking effect. The governing equations are nonlinear Schrödinger equations of Gross-Pitaevskii typewhere ǫ is the strength of the nonlinear term, µ > 0 is the nonlinearity power, and H 0 is the linear Hamiltonian with a symmetric double well potential. When µ = 1 we have a cubic nonlinearity and the resulting equation has been largely studied [9, Date: December 24, 2017.

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We consider the Gross-Petaevskii equation in 1 space dimension with a n-well trapping potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest n eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold M. Precisely, one has that solutions starting on M, or close to it, will remain close to M for times exponentially long with the inverse of the size of the nonlinearity. As heuristically expected the effective equation on M is a perturbation of a discrete nonlinear Schrödinger equation. We deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears: that is for almost all solutions starting close to M their restriction to each of the wells has norm approximatively constant over the considered time scale. In the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times. The proof is based on canonical perturbation theory; surprisingly enough, due to the Gauge invariance of the system, no non-resonance condition is required.

We consider the stationary solutions for a class of Schrödinger equations with a symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassical limit we prove that the reduction to a finitemode approximation give the stationary solutions, up to an exponentially small term, and that symmetry-breaking bifurcation occurs at a given value for the strength of the nonlinear term. The kind of bifurcation picture only depends on the non-linearity power. We then discuss the stability/instability properties of each branch of the stationary solutions. Finally, we consider an explicit one-dimensional toy model where the double well potential is given by means of a couple of attractive Dirac's delta pointwise interactions.

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