It is shown by analyzing the $1D$ Schr\"odinger equation the discontinuities in the coupling constant can occur in both the energies and the eigenfunctions. Surprisingly, those discontinuities, which are present in the energies versus the coupling constant, are of three types only: (i) discontinuous energies (similar to the 1st order phase transitions), (ii) discontinuous first derivative in the energy while the energy is continuous (similar to the 2nd order phase transitions), (ii) the energy and all its derivatives are continuous but the functions are different below and above the point of discontinuity (similar to the infinite order phase transitions). Supersymmetric (SUSY) Quantum Mechanics provides a convenient framework to study this phenomenon.

These are expanded notes of a course on basics of quantum field theory for mathematicians given by the author at MIT.

The GKSL master equation for N-level systems provides a necessary and sufficient form for the generator of a quantum dynamical semigroup in the Schrodinger picture where the underlying Hilbert space is $\mathbb{C}^N$. In this paper we provide a detailed, self-contained, and elementary construction of the GKSL master equation for an N-level system. We also provide necessary and sufficient conditions for forms of generators of semigroups which have some, but not all, of the defining properties of quantum dynamical semigroups. We do this in such a way to illuminate how each defining property of a quantum dynamical semigroup contributes to the form of the generators.

We discuss two distinct operator-theoretic settings useful for describing (or defining) propagators associated with a scalar Klein-Gordon field on a Lorentzian manifold $M$. Typically, we assume that $M$ is globally hyperbolic, but we will also consider examples where it is not. Here, the term propagator refers to any Green function or bisolution of the Klein-Gordon equation pertinent to Classical or Quantum Field Theory. These include the forward, backward, Feynman and anti-Feynman propagtors, the Pauli-Jordan function and 2-point functions of Fock states. The first operator-theoretic setting is based on the Hilbert space $L^2(M)$. This setting leads to the definition of the operator-theoretic Feynman and anti-Feynman propagators, which often (but not always) coincide with the so-called out-in Feynman and in-out anti-Feynman propagator. On some special spacetimes, the sum of the operator-theoretic Feynman and anti-Feynman propagator equals the sum of the forward and backward propagator. This is always true on static stable spacetimes and, curiously, in some other cases as well. The second setting is the Krein space $\mathcal{W}_{\rm KG}$ of solutions of the Klein-Gordon equation. Each linear operator on $\mathcal{W}_{\rm KG}$ corresponds to a bisolution of the Klein-Gordon equation, which we call its Klein-Gordon kernel. In particular, the Klein-Gordon kernels of projectors onto maximal uniformly definite subspaces are 2-point functions of Fock states, and the Klein-Gordon kernel of the identity is the Pauli-Jordan function. After a general discussion, we review a number of examples: static and asymptotically static spacetimes, FLRW spacetimes (reducible by a mode decomposition to 1-dimensional Schr\"odinger operators), deSitter space and anti-deSitter space, both proper and its universal cover.

We analyze the spectrum of the hexagonal lattice graph with a vertex coupling which manifestly violates the time reversal invariance and at high energies it asymptotically decouples edges at even degree vertices; a comparison is made to the case when such a decoupling occurs at odd degree vertices. We also show that the spectral character does not change if the equilateral elementary cell of the lattice is dilated to have three different edge lengths, except that flat bands are absent if those are incommensurate.

The goal of this paper is to settle the study of non-commutative optimal transport problems with convex regularization, in their static and finite-dimensional formulations. We consider both the balanced and unbalanced problem and show in both cases a duality result, characterizations of minimizers (for the primal) and maximizers (for the dual). An important tool we define is a non-commutative version of the classical $(c,\psi)$-transforms associated with a general convex regularization, which we employ to prove the convergence of Sinkhorn iterations in the balanced case. Finally, we show the convergence of the unbalanced transport problems towards the balanced one, as well as the convergence of transforms, as the marginal penalization parameters go to $+\infty$.

We define a path integral over Dirac operators that averages over noncommutative geometries on a fixed graph, as the title reveals, partially using quiver representations. We prove algebraic relations that are satisfied by the expectation value of the respective observables, computed in terms of integrals over unitary groups, with weights given by the spectral action. These equations generalise the Makeenko-Migdal equations -- the constraints of lattice gauge theory -- from lattices to arbitrary graphs. As a perspective, these constraints are combined with positivity conditions (on a matrix of parametrised by composition of Wilson loops). A simple example of this combination known as `bootstrap' is fully worked out.

In the present work we will give an explicit solution the problem of divergence of propgator of gauge-invariant Siegel-Zwiebach (SZ) action in Fierz-Pauli (FP) gauge (also called van Dam-Veltman-Zakharov discontinuity) by connecting it's Green's functions to that of Transverse-Traceless (TT) gauge using improved finite-field-dependent BRST (FFBRST) method.

We study the Yang-Baxter algebras $A(K,X,r)$ associated to finite set-theoretic solutions $(X,r)$ of the braid relations. We introduce an equivalent set of quadratic relations $\Re\subseteq G$, where $G$ is the reduced Gr\"obner basis of $(\Re)$. We show that if $(X,r)$ is left-nondegenerate and idempotent then $\Re= G$ and the Yang-Baxter algebra is PBW. We use graphical methods to study the global dimension of PBW algebras in the $n$-generated case and apply this to Yang-Baxter algebras in the left-nondegenerate idempotent case. We study the $d$-Veronese subalgebras for a class of quadratic algebras and use this to show that for $(X,r)$ left-nondegenerate idempotent, the $d$-Veronese subalgebra $A(K,X,r)^{(d)}$ can be identified with $A(K,X,r^{(d)})$, where $(X,r^{(d)})$ are all left-nondegenerate idempotent solutions. We determined the Segre product in the left-nondegenerate idempotent setting. Our results apply to a previously studied class of `permutation idempotent' solutions, where we show that all their Yang-Baxter algebras for a given cardinality of $X$ are isomorphic and are isomorphic to their $d$-Veronese subalgebras. In the linearised setting, we construct the Koszul dual of the Yang-Baxter algebra and the Nichols-Woronowicz algebra in the idempotent case, showing that the latter is quadratic. We also construct noncommutative differentials on some of these quadratic algebras.

In this study, we conduct parameter estimation analysis on a data assimilation algorithm for two turbulence models: the simplified Bardina model and the Navier-Stokes-{\alpha} model. Our approach involves creating an approximate solution for the turbulence models by employing an interpolant operator based on the observational data of the systems. The estimation depends on the parameter alpha in the models. Additionally, numerical simulations are presented to validate our theoretical results

The study of symmetries in the realm of manifolds can be approached in two different ways. On one hand, Killing vector fields on a (pseudo-)Riemannian manifold correspond to the directions of local isometries within it. On the other hand, from an algebraic perspective, global symmetries of such manifolds are associated with group elements. Although the connection between these two concepts is well established in the literature, this work aims to build an unexplored bridge between the Killing vector fields of n-dimensional maximally symmetric spaces and their corresponding isometry Lie groups, anchoring primarily on the definition of induced vector fields. As an application of our main result, we explore two specific examples: the three-dimensional Euclidean space and Minkowski spacetime.

Through the rotational invariance of the 2-d critical bond percolation exploration path on the square lattice we express Smirnov's edge parafermionic observable as a sum of two new edge observables. With the help of these two new edge observables we can apply the discrete harmonic analysis and conformal mapping theory to prove the convergence of the 2-d critical bond percolation exploration path on the square lattice to the trace of $SLE_6$ as the mesh size of the lattice tends to zero.

One of the fundamental problems in quantum mechanics is finding the correct quantum image of a classical observable that would correspond to experimental measurements. We investigate for the appropriate quantization rule that would yield a Hamiltonian that obeys the quantum analogue of Hamilton's equations of motion, which includes differentiation of operators with respect to another operator. To give meaning to this type of differentiation, Born and Jordan established two definitions called the differential quotients of first type and second type. In this paper we modify the definition for the differential quotient of first type and establish its consistency with the differential quotient of second type for different basis operators corresponding to different quantizations. Theorems and differentiation rules including differentiation of operators with negative powers and multiple differentiation were also investigated. We show that the Hamiltonian obtained from Weyl, simplest symmetric, and Born-Jordan quantization all satisfy the required algebra of the quantum equations of motion.

In this paper, we develop a mathematical framework for generating strong customized field concentration locally around the inhomogeneous medium inclusion via surface transmission resonance. The purpose of this paper is twofold. Firstly, we show that for a given inclusion embedded in an otherwise uniformly homogeneous background space, we can design an incident field to generate strong localized field concentration at any specified places around the inclusion. The aforementioned customized field concentration is crucially reliant on the peculiar spectral and geometric patterns of certain transmission eigenfunctions. Secondly, we prove the existence of a sequence of transmission eigenfunctions for a specific wavenumber and they exhibit distinct surface resonant behaviors, accompanying strong surface-localization and surface-oscillation properties. These eigenfunctions as the surface transmission resonant modes fulfill the requirement for generating the field concentration.

For a given region, and specified boundary flux and density rate of an extensive property, the optimal flux field that satisfies the balance conditions is considered. The optimization criteria are the $L^{p}$-norm and a Sobolev-like norm of the flux field. Finally, the capacity of the region to accommodate various boundary fluxes and density rates is defined and analyzed.

In this project, we provide a deep-learning neural network (DNN) based biophysics model to predict protein properties. The model uses multi-scale and uniform topological and electrostatic features generated with protein structural information and force field, which governs the molecular mechanics. The topological features are generated using the element specified persistent homology (ESPH) while the electrostatic features are fast computed using a Cartesian treecode. These features are uniform in number for proteins with various sizes thus the broadly available protein structure database can be used in training the network. These features are also multi-scale thus the resolution and computational cost can be balanced by the users. The machine learning simulation on over 4000 protein structures shows the efficiency and fidelity of these features in representing the protein structure and force field for the predication of their biophysical properties such as electrostatic solvation energy. Tests on topological or electrostatic features alone and the combination of both showed the optimal performance when both features are used. This model shows its potential as a general tool in assisting biophysical properties and function prediction for the broad biomolecules using data from both theoretical computing and experiments.

We begin with an exact expression for the entropy of a system of hard spheres within the Hamming space. This entropy relies on probability marginals, which are determined by an extended set of Belief Propagation (BP) equations. The BP probability marginals are functions of auxiliary variables which are introduced to model the effects of loopy interactions on a tree-structured interaction graph. We explore various reasonable and approximate probability distributions, ensuring they align with the exact solutions of the BP equations. Our approach is based on an ansatz of (in)homogeneous cavity marginals respecting the permutation symmetry of the problem. Through thorough analysis, we aim to minimize errors in the BP equations. Our findings support the conjecture that the maximum packing density asymptotically conforms to the lower bound proposed by Gilbert and Varshamov, further validated by the solution of the loopy BP equations.

We consider the derivative of the characteristic polynomial of $N \times N$ Haar distributed unitary matrices. In the limit $N \to \infty$, we give a formula for general non-integer moments of the derivative for values of the spectral variable inside the unit disc. The formula we obtain is given in terms of a confluent hypergeometric function. We give an extension of this result to joint moments taken at different points, again valid for non-integer moment orders. For integer moments, our results simplify and are expressed as a single Laguerre polynomial. Using these moment formulae, we derive the mean counting function of zeros of the derivative in the limit $N \to \infty$. This motivated us to consider related questions for the derivative of the Riemann zeta function. Assuming the Lindel\"of hypothesis, we show that integer moments away from but approaching the critical line agree with our random matrix results. Within random matrix theory, we obtain an explicit expression for the integer moments, for finite matrix size, as a sum over partitions. We use this to obtain asymptotic formulae in a regime where the spectral variable approaches or is on the unit circle.