Abstract:In this paper we analyze a free quantum particle in a straight Dirichlet waveguide which has at its axis two Dirichlet barriers of lengths ℓ ± separated by a window of length 2a. It is known that if the barriers are semiinfinite, i.e. we have two adjacent waveguides coupled laterally through the boundary window, the system has for any a > 0 a finite number of eigenvalues below the essential spectrum threshold. Here we demonstrate that for large but finite ℓ ± the system has resonances which converge to the said eigenvalues as ℓ ± → ∞, and derive the leading term in the corresponding asymptotic expansion.
A periodic selfadjoint differential operator of even order and with distant perturbations in a multidimensional space is treated. The role of perturbations is played by arbitrary localized operators. The localization is described by specially chosen weight functions. The behavior of the spectrum of the perturbed operator is studied under the condition that the distance between the domains where the perturbation are localized tends to infinity. It is shown that there exists a simple isolated eigenvalue of the perturbed operator that tends to a simple isolated eigenvalue of the limit operator. Series expansions are obtained for this eigenvalue of the perturbed operator and for the corresponding eigenfunction. Uniform convergence for these series is shown and formulas for their terms are deduced.2010 Mathematics Subject Classification. Primary 35B20.
Ubiquitin-specific protease 7 (USP7) regulates the stability of a plethora of intracellular proteins involved in the suppression of anti-tumor immune responses and its overexpression is associated with poor survival in many cancers. USP7 impairs the balance of the p53/MDM2 axis resulting in the proteasomal degradation of the p53 tumor suppressor, a process that can be reversed by small-molecule inhibitors of USP7. USP7 was shown to regulate the anti-tumor immune responses in several cases. Its inhibition impedes the function of regulatory T cells, promotes polarization of tumor-associated macrophages, and reduces programmed death-ligand 1 (PD‑L1) expression in tumor cells. The efficacy of small-molecule USP7 inhibitors was demonstrated in vivo. The synergistic effect of combining USP7 inhibition with cancer immunotherapy is a promising therapeutic approach, though its clinical efficacy is yet to be proven. In this review, we focus on the recent developments in understanding the intrinsic role of USP7, its interplay with other molecular pathways, and the therapeutic potential of targeting USP7 functions.
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