Perturbation theory of PT-symmetric Hamiltonians

In the framework of perturbation theory the reality of the perturbed eigenvalues of a class of \PTsymmetric Hamiltonians is proved using stability techniques. We apply this method to \PTsymmetric unperturbed Hamiltonians perturbed by \PTsymmetric additional interactions

Canonical Expansion of PT-Symmetric Operators and Perturbation Theory

Let $H$ be any \PT symmetric Schr\"odinger operator of the type $-\hbar^2\Delta+(x_1^2+...+x_d^2)+igW(x_1,...,x_d)$ on $L^2(\R^d)$, where $W$ is any odd homogeneous polynomial and $g\in\R$. It is proved that $\P H$ is self-adjoint and that its eigenvalues coincide (up to a sign) with the singular values of $H$, i.e. the eigenvalues of $\sqrt{H^\ast H}$. Moreover we explicitly construct the canonical expansion of $H$ and determine the singular values $\mu_j$ of $H$ through the Borel summability of their divergent perturbation theory. The singular values yield estimates of the location of the eigenvalues \l_j of $H$ by Weyl's inequalities.Comment: 20 page

PTPT symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum

Consider in $L^2(R^d)$, $d\geq 1$, the operator family $H(g):=H_0+igW$. \ds H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2 is the quantum harmonic oscillator with rational frequencies, $W$ a $P$ symmetric bounded potential, and $g$ a real coupling constant. We show that if $|g|<\rho$, $\rho$ being an explicitly determined constant, the spectrum of $H(g)$ is real and discrete. Moreover we show that the operator \ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1 has real discrete spectrum but is not diagonalizable.Comment: 20 page

Recent Advances in Understanding the Protein Corona of Nanoparticles and in the Formulation of “Stealthy” Nanomaterials

In the last decades, the staggering progress in nanotechnology brought around a wide and heterogeneous range of nanoparticle-based platforms for the diagnosis and treatment of many diseases. Most of these systems are designed to be administered intravenously. This administration route allows the nanoparticles (NPs) to widely distribute in the body and reach deep organs without invasive techniques. When these nanovectors encounter the biological environment of systemic circulation, a dynamic interplay occurs between the circulating proteins and the NPs, themselves. The set of proteins that bind to the NP surface is referred to as the protein corona (PC). PC has a critical role in making the particles easily recognized by the innate immune system, causing their quick clearance by phagocytic cells located in organs such as the lungs, liver, and spleen. For the same reason, PC defines the immunogenicity of NPs by priming the immune response to them and, ultimately, their immunological toxicity. Furthermore, the protein corona can cause the physical destabilization and agglomeration of particles. These problems induced to consider the PC only as a biological barrier to overcome in order to achieve efficient NP-based targeting. This review will discuss the latest advances in the characterization of PC, development of stealthy NP formulations, as well as the manipulation and employment of PC as an alternative resource for prolonging NP half-life, as well as its use in diagnostic applications

New Insights into Bile Acids Related Signaling Pathways in the Onset of Colorectal Cancer

Colorectal cancer (CRC) ranks as the second among the causes of tumor death worldwide, with an estimation of 1.9 million new cases in 2020 and more than 900,000 deaths. This rate might increase by 60% over the next 10 years. These data are unacceptable considering that CRC could be successfully treated if diagnosed in the early stages. A high-fat diet promotes the hepatic synthesis of bile acids (BAs) increasing their delivery to the colonic lumen and numerous scientific reports correlate BAs, especially secondary BAs, with CRC incidence. We reviewed the physicochemical and biological characteristics of BAs, focusing on the major pathways involved in CRC risk and progression. We specifically pointed out the role of BAs as signaling molecules and the tangled relationships among their nuclear and membrane receptors with the big bang of molecular and cellular events that trigger CRC occurrence

Agri-Food Waste from Apple, Pear, and Sugar Beet as a Source of Protective Bioactive Molecules for Endothelial Dysfunction and Its Major Complications

Endothelial damage is recognized as the initial step that precedes several cardiovascular diseases (CVD), such as atherosclerosis, hypertension, and coronary artery disease. It has been demonstrated that the best treatment for CVD is prevention, and, in the frame of a healthy lifestyle, the consumption of vegetables, rich in bioactive molecules, appears effective at reducing the risk of CVD. In this context, the large amount of agri-food industry waste, considered a global problem due to its environmental and economic impact, represents an unexplored source of bioactive compounds. This review provides a summary regarding the possible exploitation of waste or by-products derived by the processing of three traditional Italian crops—apple, pear, and sugar beet—as a source of bioactive molecules to protect endothelial function. Particular attention has been given to the bioactive chemical profile of these pomaces and their efficacy in various pathological conditions related to endothelial dysfunction. The waste matrices of apple, pear, and sugar beet crops can represent promising starting material for producing “upcycled” products with functional applications, such as the prevention of endothelial dysfunction linked to cardiovascular diseases

On the eigenproblems of PT-symmetric oscillators

We consider the non-Hermitian Hamiltonian H= -\frac{d^2}{dx^2}+P(x^2)-(ix)^{2n+1} on the real line, where P(x) is a polynomial of degree at most n \geq 1 with all nonnegative real coefficients (possibly P\equiv 0). It is proved that the eigenvalues \lambda must be in the sector | arg \lambda | \leq \frac{\pi}{2n+3}. Also for the case H=-\frac{d^2}{dx^2}-(ix)^3, we establish a zero-free region of the eigenfunction u and its derivative u^\prime and we find some other interesting properties of eigenfunctions.Comment: 21pages, 9 figure

Treatment with PCSK9 Inhibitor Evolocumab Improves Vascular Oxidative Stress and Arterial Stiffness in Hypercholesterolemic Patients with High Cardiovascular Risk

Atherosclerosis and atherosclerotic-related cardiovascular diseases (ASCVD) are characterized by high serum levels of low-density lipoprotein cholesterol (LDL-C) that can promote the generation of reactive oxygen species (ROS). To answer the need for better LDL-C control in individuals at high and very high risk for CVD, a new injectable innovative family of lipid-lowering (LL) monoclonal antibodies against the protein convertase subtilisin/kexin type 9 (PCSK9) has been approved. However, the effect of these drugs on vascular function, such as ROS generation and arterial stiffness, has not already been extensively described. In this report, we present data from 18 males with high to very high CV risk undergoing LL treatment (LLT) with either statin and ezetimibe or ezetimibe monotherapy, who experienced, after a 2-month treatment with Evolocumab, a significant improvement in blood pressure (BP)-adjusted carotid-femoral pulse wave velocity (cfPWV) (p-value = 0.0005 in the whole cohort, p-value = 0.0046 in the sub-cohort undergoing background LLT with statin and ezetimibe, p-value = 0.015 in the sub-cohort undergoing background LLT with ezetimibe monotherapy), which was significantly associated with a decrease in freshly isolated leukocytes (PBMCS)-derived H2O2 production (p-value = 0.004, p-value = 0.02 and p-value = 0.05, respectively, in the whole cohort, in the statin + ezetimibe sub-cohort, and the ezetimibe sub-cohort). Our observations support the role of systemic oxidative stress in atherosclerosis and give a further rationale for using Evolocumab also for its effect in vascular disorders linked to oxidative processes

Eigenvalues of PT-symmetric oscillators with polynomial potentials

We study the eigenvalue problem $-u^{\prime\prime}(z)-[(iz)^m+P_{m-1}(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\frac{\pi}{2}\pm \frac{2\pi}{m+2}$, where $P_{m-1}(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial and integers $m\geq 3$. We provide an asymptotic expansion of the eigenvalues $\lambda_n$ as $n\to+\infty$, and prove that for each {\it real} polynomial $P_{m-1}$, the eigenvalues are all real and positive, with only finitely many exceptions.Comment: 23 pages, 1 figure. v2: equation (14) as well as a few subsequent equations has been changed. v3: typos correcte