In this paper, it is shown that a one-dimensional Hamiltonian with an attractive delta potential at the origin plus a mass jump at the same point cannot have a bound state, as is the case with the ordinary attractive delta potential with constant mass, unless another term is added into the potential in the form of a derivative of the Dirac delta at the origin. The consistency of this singular potential with two terms is guaranteed by choosing suitable matching conditions at the singular point for the wavefunctions. Furthermore, it is proved that the self-adjointness of the Hamiltonian with both singular interactions determines the coefficient of the derivative of the delta in a unique manner. Under these conditions, the bound state and its energy are obtained and it is checked that the correct results in the limit of equal masses are obtained.

No abstract

Gaze stabilization reflexes reduce motion blur and simplify the processing of visual information by keeping the eyes level. These reflexes typically depend on estimates of the rotational motion of the body, head, and eyes, acquired by visual or mechanosensory systems. During rapid movements, there can be insufficient time for sensory feedback systems to estimate rotational motion, and additional mechanisms are required. The solutions to this common problem likely reflect an animal's behavioral repertoire. Here, we examine gaze stabilization in three families of dipteran flies, each with distinctly different flight behaviors. Through frequency response analysis based on tethered-flight experiments, we demonstrate that fast roll oscillations of the body lead to a stable gaze in hoverflies, whereas the reflex breaks down at the same speeds in blowflies and horseflies. Surprisingly, the high-speed gaze stabilization of hoverflies does not require sensory input from the halteres, their low-latency balance organs. Instead, we show how the behavior is explained by a hybrid control system that combines a sensory-driven, active stabilization component mediated by neck muscles, and a passive component which exploits physical properties of the animal's anatomy---the mass and inertia of the head. This solution requires hoverflies to have specializations of the head-neck joint that can be employed during flight. Our comparative study highlights how species-specific control strategies have evolved to support different visually-guided flight behaviors.