On Exactness Of The Supersymmetric WKB Approximation Scheme

Exactness of the lowest order supersymmetric WKB (SWKB) quantization condition $\int^{x_2}_{x_1} \sqrt{E-\omega^2(x)} dx = n \hbar \pi$, for certain potentials, is examined, using complex integration technique. Comparison of the above scheme with a similar, but {\it exact} quantization condition, $\oint_c p(x,E) dx = 2\pi n \hbar$, originating from the quantum Hamilton-Jacobi formalism reveals that, the locations and the residues of the poles that contribute to these integrals match identically, for both of these cases. As these poles completely determine the eigenvalues in these two cases, the exactness of the SWKB for these potentials is accounted for. Three non-exact cases are also analysed; the origin of this non-exactness is shown to be due the presence of additional singularities in $\sqrt{E-\omega^2(x)}$, like branch cuts in the $x-$plane.Comment: 11 pages, latex, 1 figure available on reques

Periodic Quasi - Exactly Solvable Models

Various quasi-exact solvability conditions, involving the parameters of the periodic associated Lam{\'e} potential, are shown to emerge naturally in the quantum Hamilton-Jacobi approach. It is found that, the intrinsic nonlinearity of the Riccati type quantum Hamilton-Jacobi equation is primarily responsible for the surprisingly large number of allowed solvability conditions in the associated Lam{\'e} case. We also study the singularity structure of the quantum momentum function, which yields the band edge eigenvalues and eigenfunctions.Comment: 11 pages, 5 table

Quantum Hamilton-Jacobi analysis of PT symmetric Hamiltonians

We apply the quantum Hamilton-Jacobi formalism, naturally defined in the complex domain, to a number of complex Hamiltonians, characterized by discrete parity and time reversal (PT) symmetries and obtain their eigenvalues and eigenfunctions. Examples of both quasi-exactly and exactly solvable potentials are analyzed and the subtle differences, in the singularity structures of their quantum momentum functions, are pointed out. The role of the PT symmetry in the complex domain is also illustrated.Comment: 11 page

Gene expression and data analysis pipeline using cancer BioPortal in the classroom

At institutions with an emphasis on authentic research experiences as an integral part of the biology curriculum, COVID created a huge challenge for course instructors whose learning objectives were designed for such experiences. Moving such laboratory experiences online when remote learning became necessary has resulted in a new model for CUREs that utilizes free online databases to provide not only a novel research experience for students, but also the opportunity to engage in big data analysis. Cancer BioPortal (cBioPortal) is an open-access collective cancer research resource for storing and exploring clinical, genomic, proteomic, and transcriptomic data. cBioPortal eliminates the computational barrier of interpreting complex genomic data by providing easily understandable visualization that can be interpreted and translated into relevant biological insights. Because no prior computational knowledge is required, cBioPortal is an ideal educational tool for either in-person or distance learning environments. We developed a pedagogical approach, video tutorials, and data analysis workflows centered on using cBioPortal. Pedagogically, students develop an initial research outline that is continually updated and graded throughout the project. Progress during the project or course is assessed by a series of student presentations that are 5 to 15 minutes in length and are aimed at explaining the approach used in data acquisition, interpretation of the data, and relevance to the initial hypothesis. While cancer-specific, this analysis platform appeals to a wide range of classes and student interests. Further, the project has been successfully done both as an independent research experience and as part of a virtual class-based research project

Calculation of Band Edge Eigenfunctions and Eigenvalues of Periodic Potentials through the Quantum Hamilton - Jacobi Formalism

We obtain the band edge eigenfunctions and the eigenvalues of solvable periodic potentials using the quantum Hamilton - Jacobi formalism. The potentials studied here are the Lam{\'e} and the associated Lam{\'e} which belong to the class of elliptic potentials. The formalism requires an assumption about the singularity structure of the quantum momentum function $p$, which satisfies the Riccati type quantum Hamilton - Jacobi equation, $p^{2} -i \hbar \frac{d}{dx}p = 2m(E- V(x))$ in the complex $x$ plane. Essential use is made of suitable conformal transformations, which leads to the eigenvalues and the eigenfunctions corresponding to the band edges in a simple and straightforward manner. Our study reveals interesting features about the singularity structure of $p$, responsible in yielding the band edge eigenfunctions and eigenvalues.Comment: 21 pages, 5 table

Quantum interference within the complex quantum Hamilton-Jacobi formalism

Quantum interference is investigated within the complex quantum Hamilton-Jacobi formalism. As shown in a previous work [Phys. Rev. Lett. 102, 250401 (2009)], complex quantum trajectories display helical wrapping around stagnation tubes and hyperbolic deflection near vortical tubes, these structures being prominent features of quantum caves in space-time Argand plots. Here, we further analyze the divergence and vorticity of the quantum momentum function along streamlines near poles, showing the intricacy of the complex dynamics. Nevertheless, despite this behavior, we show that the appearance of the well-known interference features (on the real axis) can be easily understood in terms of the rotation of the nodal line in the complex plane. This offers a unified description of interference as well as an elegant and practical method to compute the lifetime for interference features, defined in terms of the average wrapping time, i.e., considering such features as a resonant process.Comment: revised version, 13 pages, 11 figures, 1 tabl

Disruption of a structurally important extracellular element in the Glycine Receptor leads to decreased synaptic integration and signaling resulting in Severe Startle Disease

Functional impairments or trafficking defects of inhibitory glycine receptors (GlyRs) have been linked to human hyperekplexia/startle disease and autism spectrum disorders. We found that a lack of synaptic integration of GlyRs, together with disrupted receptor function, is responsible for a lethal startle phenotype in a novel spontaneous mouse mutant shaky, caused by a missense mutation, Q177K, located in the extracellular β8–β9 loop of the GlyR α1 subunit. Recently, structural data provided evidence that the flexibility of the β8–β9 loop is crucial for conformational transitions during opening and closing of the ion channel and represents a novel allosteric binding site in Cys-loop receptors. We identified the underlying neuropathological mechanisms in male and female shaky mice through a combination of protein biochemistry, immunocytochemistry, and both in vivo and in vitro electrophysiology. Increased expression of the mutant GlyR α1Q177K subunit in vivo was not sufficient to compensate for a decrease in synaptic integration of α1Q177Kβ GlyRs. The remaining synaptic heteromeric α1Q177Kβ GlyRs had decreased current amplitudes with significantly faster decay times. This functional disruption reveals an important role for the GlyR α1 subunit β8–β9 loop in initiating rearrangements within the extracellular–transmembrane GlyR interface and that this structural element is vital for inhibitory GlyR function, signaling, and synaptic clustering

Accuracy of Semiclassical Methods for Shape Invariant Potentials

We study the accuracy of several alternative semiclassical methods by computing analytically the energy levels for many large classes of exactly solvable shape invariant potentials. For these potentials, the ground state energies computed via the WKB method typically deviate from the exact results by about 10%, a recently suggested modification using nonintegral Maslov indices is substantially better, and the supersymmetric WKB quantization method gives exact answers for all energy levels.Comment: 7 pages, Latex, and two tables in postscrip