The Effects on Employees from the Switch to Mandatory Contributions in the University of Arkansas Retirement Plan

After the 2016 fiscal year, the University of Arkansas Retirement Plan instituted mandatory contributions for full-time employees, presumably to boost retirement savings among those least prepared for retirement. Mandatory contributions began at 1% in fiscal-year 2017 and increased annually to 5% in fiscal-year 2022. This change may have harmed employees with tight budget constraints who wish to contribute less than the minimum contribution rate. At the same time, it may have helped those who were saving less than their optimal amount due to behavioral biases. We surveyed employees at the University of Arkansas campus to assess the effects from the change to mandatory contributions and received 171 responses. Our main findings are that most respondents are unaffected by the change to mandatory contributions; a small minority are unsatisfied with the change; average contribution rates increased for all full-time employees, especially staff; a small percentage of staff, but no faculty, may have been harmed by the change; and a larger percentage of staff and faculty may have been helped. These results, however, must be interpreted with caution because they are limited by a relatively small sample size that is not representative of the employee composition at the University of Arkansas. For more robust results, a much larger survey is needed that reaches across all campuses of the University of Arkansas System to accurately assess the effects on employees from mandatory contributions

An isoperimetric problem for leaky loops and related mean-chord inequalities

We consider a class of Hamiltonians in $L^2(\R^2)$ with attractive interaction supported by piecewise $C^2$ smooth loops $\Gamma$ of a fixed length $L$, formally given by $-\Delta-\alpha\delta(x-\Gamma)$ with $\alpha>0$. It is shown that the ground state of this operator is locally maximized by a circular $\Gamma$. We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of $\Gamma$.Comment: LaTeX, 16 page

Leaky quantum graphs: approximations by point interaction Hamiltonians

We prove an approximation result showing how operators of the type $-\Delta -\gamma \delta (x-\Gamma)$ in $L^2(\mathbb{R}^2)$, where $\Gamma$ is a graph, can be modeled in the strong resolvent sense by point-interaction Hamiltonians with an appropriate arrangement of the $\delta$ potentials. The result is illustrated on finding the spectral properties in cases when $\Gamma$ is a ring or a star. Furthermore, we use this method to indicate that scattering on an infinite curve $\Gamma$ which is locally close to a loop shape or has multiple bends may exhibit resonances due to quantum tunneling or repeated reflections.Comment: LaTeX 2e, 31 pages with 18 postscript figure

On the discrete spectrum of spin-orbit Hamiltonians with singular interactions

We give a variational proof of the existence of infinitely many bound states below the continuous spectrum for spin-orbit Hamiltonians (including the Rashba and Dresselhaus cases) perturbed by measure potentials thus extending the results of J.Bruening, V.Geyler, K.Pankrashkin: J. Phys. A 40 (2007) F113--F117.Comment: 10 pages; to appear in Russian Journal of Mathematical Physics (memorial volume in honor of Vladimir Geyler). Results improved in this versio

Distribution theory for Schr\"odinger's integral equation

Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schr\"odinger's equation. This paper, in contrast, investigates the integral form of Schr\"odinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schr\"odinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schr\"odinger's differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov's result to hypersurfaces. Second, we derive a new closed-form solution to Schr\"odinger's integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schr\"odinger's differential equation. Third, we derive boundary conditions for `super-singular' potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution, and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schr\"odinger's integral equation is viable tool for studying singular interactions in quantum mechanics.Comment: 23 page

Scattering by local deformations of a straight leaky wire

We consider a model of a leaky quantum wire with the Hamiltonian $-\Delta -\alpha \delta(x-\Gamma)$ in $L^2(\R^2)$, where $\Gamma$ is a compact deformation of a straight line. The existence of wave operators is proven and the S-matrix is found for the negative part of the spectrum. Moreover, we conjecture that the scattering at negative energies becomes asymptotically purely one-dimensional, being determined by the local geometry in the leading order, if $\Gamma$ is a smooth curve and $\alpha \to\infty$.Comment: Latex2e, 15 page

Bound states in point-interaction star-graphs

We discuss the discrete spectrum of the Hamiltonian describing a two-dimensional quantum particle interacting with an infinite family of point interactions. We suppose that the latter are arranged into a star-shaped graph with N arms and a fixed spacing between the interaction sites. We prove that the essential spectrum of this system is the same as that of the infinite straight "polymer", but in addition there are isolated eigenvalues unless N=2 and the graph is a straight line. We also show that the system has many strongly bound states if at least one of the angles between the star arms is small enough. Examples of eigenfunctions and eigenvalues are computed numerically.Comment: 17 pages, LaTeX 2e with 9 eps figure