The spin-1/2 Aharonov-Bohm problem is examined in the Galilean limit for the case in which a Coulomb potential is included. It is found that the application of the selfadjoint extension method to this system yields singular solutions only for one-half the full range of flux parameter which is allowed in the limit of vanishing Coulomb potential. Thus one has a remarkable example of a case in which the condition of normalizability is necessary but not sufficient for the occurrence of singular solutions. Expressions for the bound state energies are derived. Also the conditions for the occurrence of singular solutions are obtained when the non-gauge potential is ξ/r p (0 ≤ p < 2).
Schizophrenia is a psychiatric disorder that affects over 20 million people globally. Notably, schizophrenia is associated with decreased density of dendritic spines and decreased levels of D-serine, a co-agonist of the N-methyl-D-aspartate receptor (NMDAR), and hypofunction of NMDARs is thought to play a role in the pathophysiology of schizophrenia. We hypothesized that lowered D-serine levels associated with schizophrenia would bias toward ion flux-independent signaling by the NMDAR, which drives spine shrinkage and loss. Using a schizophrenia mouse model lacking the enzyme for D-serine production (serine racemase knock out; SRKO), we show that activity-dependent spine growth is inhibited in SRKO mice but can be acutely rescued by exogenous D-serine. When examining a wider range of stimulus strengths, we observed activity-dependent spine growth at higher stimulus strengths, but overall found a strong bias toward spine shrinkage in the SRKO mice as compared to wild-type littermates. Furthermore, we demonstrate that enhanced ion flux-independent signaling through the NMDAR contributes to this bias toward spine shrinkage, which is likely exacerbated by an increase in synaptic NMDARs in hippocampal synapses of SRKO mice. Our results support a model in which the lowered D-serine levels associated with schizophrenia lead to increased ion flux-independent NMDAR signaling and a bias toward spine shrinkage that could play an important role in the loss of dendritic spines associated with schizophrenia.
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