We prove a reverse Lieb--Thirring inequality with a sharp constant for the matrix Schr\"odinger equation on the half-line.

We consider the Euclidean vacuum for linearized gravity on the global de Sitter space, obtained from the Euclidean Green's function on the 4-sphere. We use the notion of Calder\'on projectors to recover a quantum state for the Lorentzian theory on de Sitter space. We show that while the state is gauge invariant and Hadamard, it is not positive on the whole of the phase space. We show however that a suitable modification at low energies yields a well-defined Hadamard state on global de Sitter space.

We investigate the continuous extension of discrete shift translations on one dimensional quantum lattice systems. We focus on a specific construction provided by a quasi-free C*-flow on the one-dimensional fermion lattice system. This quasi-free C*-flow is heuristically generated by a long-range Hamiltonian consisting of two-body translation invariant interactions with $\frac{1}{r}$-decay. We explore statistical-mechanical interpretation of such a long-range Hamiltonian, which may be more naturally associated with a momentum operator rather than the Hamiltonian. Through its explicit dynamical formulas, wherein notably the sinc function appears, the continuous shift translations reveal violations of causality and locality. Furthermore, we demonstrate that this quasi-free C*-flow, implementing the shift translations, cannot be extended to the one-dimensional quantum spin system via the Jordan-Wigner transformation.

A new algebra, hitherto not encountered in the usual Lie algebraic varieties or supervarieties, is introduced. The paper explores the rich and novel structure of the algebra, and it compares it on the one hand with the Jordan-Lie Superalgebras studied by Okubo and Kamiya, and on the other, with the four usual Euclidean division rings of the reals, the complexes, the quaternions and the octonions that the algebra is seen to combine, extend and generalise. A potential physical application of the algebra is briefly alluded to at the end.

This work investigates the intricate relationship between the q-boson model, a quantum integrable system, and classical integrable systems such as the Toda and KP hierarchies. Initially, we analyze scalar products of off-shell Bethe states and explore their connections to tau functions of integrable hierarchies. Furthermore, we discuss correlation functions within this formalism, examining their representations in terms of tau functions, as well as their Schur polynomial expansions.

We prove a rigorous lower bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of singular interactions, including the Coulomb potential. Combined with the upper bound obtained in \cite{ChrHaiNam-23b}, our result establishes an analogue of the Gell-Mann--Brueckner formula $c_{1}\rho\log\left(\rho\right)+c_{2}\rho$ for the correlation energy of the electron gas in the high-density limit. Moreover, our analysis allows us to go beyond mean-field scaling while still covering the same class of potentials.

We develop a Schwinger-Keldysh field theory (SKFT) for open quantum systems interacting with a dissipative environment and apply it to the spin-boson model as an archetypical example where the environment is composed of a bosonic bath. Prior SKFT developments of this type have been confined to the Markovian regime, as an alternative to a conventional description by the Lindblad quantum master equation (QME) which is a time-local matrix differential equation. Here we combine SKFT with a two-particle irreducible (2PI) action that resums a class of Feynman diagrams to infinite order. We obtain the time-evolution of the spin density matrix in the form of a system of integro-differential equations applicable to both Markovian and non-Markovian regimes. The latter regime--where taking into account memory effects becomes essential--poses a challenge for standard methods when trying to incorporate arbitrary properties of the system, bath, and length of time evolution. The SKFT+2PI-computed time evolution of the spin expectation values in the Markovian regime reproduces the solution of the Lindblad QME, as long as the system-bath coupling in the latter is adjusted by increasing it. In the non-Markovian regime, SKFT+2PI yields a nonperturbative solution that mimics results from both hierarchical equations of motion and tensor networks methods that we employ as benchmarks. Our SKFT+2PI approach can also access challenging cases, such as zero-temperature and sub-Ohmic bath, as well as arbitrary long evolution times. Taking into account favorable numerical cost of solving the integro-differential equations with increasing number of spins, time steps or dimensionality the SKFT+2PI approach offers a promising route for simulation of driven-dissipative systems in quantum computing or quantum magnonics and spintronics in the presence of a variety of (single or multiple) dissipative environments.

Finsler geometry is a natural generalization of (pseudo-)Riemannian geometry, where the line element is not the square root of a quadratic form but a more general homogeneous function. Parameterizing this in terms of symmetric tensors suggests a possible interpretation in terms of higher-spin fields. We will see here that, at linear level in these fields, the Finsler version of the Ricci tensor leads to the curved-space Fronsdal equation for all spins, plus a Stueckelberg-like coupling. Nonlinear terms can also be systematically analyzed, suggesting a possible interacting structure. No particular choice of spacetime dimension is needed. The Stueckelberg mechanism breaks gauge transformations to a redundancy that does not change the geometry. This is however not enough to eliminate non-transverse modes, at least for some versions of Finsler dynamics.

In this paper, we present a controllability analysis of the quantum Ising periodic chain of n spin 1/2 particles where the interpolating parameter between the two Hamiltonians plays the role of the control. A fundamental result in the control theory of quantum systems states that the set of achievable evolutions is (dense in) the Lie group corresponding to the Lie algebra generated by the Hamiltonians of the system. Such a dynamical Lie algebra therefore characterizes all the state transitions available for a given system. For the Ising spin periodic chain we characterize such a dynamical Lie algebra and therefore the set of all reachable states. In particular, we prove that the dynamical Lie algebra is a (3n-1)-dimensional Lie sub-algebra of su(2^n) which is a direct sum of a two dimensional center and a (3n-3)-dimensional semisimple Lie subalgebra. This in turn is the direct sum of n-1 Lie algebras isomorphic to su(2) parametrized by the eigenvalues of a fixed matrix. We display the basis for each of these Lie subalgebras. Therefore the problem of control for the Ising spin periodic chain is, modulo the two dimensional center, a problem of simultaneous control of n-1 spin 1/2 particles. In the process of proving this result, we develop some tools which are of general interest for the controllability analysis of quantum systems with symmetry.

We study the point spectrum of a second order difference operator with complex potential on the half-line via Fredholm determinants of the corresponding Birman-Schwinger operator pencils, the Evans and the Jost functions. An application is given to instability of a generalization of the Kolmogorov flow for the Euler equation of ideal fluid on the two dimensional torus.

We numerically analyse solutions of the spherically symmetric gravitational Vlasov-Poisson system close to compactly supported stable steady states. We observe either partially undamped oscillations or macroscopically damped solutions. We investigate for many steady states to which of these behaviours they correspond. A linear relation between the exponents of polytropic steady states and the qualitative behaviour close to them is identified. Undamped oscillations are also observed around not too concentrated King models and around all shells with a sufficiently large vacuum region. We analyse all solutions both at the non-linear and linearised level and find that the qualitative behaviours are identical at both. To relate the observed phenomena to theoretical results, we further include a comprehensive numerical study of the radial particle periods in the equilibria.

We extend the port-Hamiltonian framework defined with respect to a Lagrangian submanifold and a Dirac structure by augmenting the Lagrangian submanifold with the space of external variables. The new pair of conjugated variables is called energy port. We show that in the most general case, the extension describes constrained Hamiltonian systems whose Hamiltonian function depends on inputs.

Using an exact holographic duality formula between the inhomogeneous 2d Ising model and 3d quantum gravity, we provide a formula for "real" zeroes of the 2d Ising partition function on finite graphs in terms of the geometry of a 2d triangulation embedded in the three-dimensional Euclidean space. The complex phase of those zeroes is given by the dihedral angles of the triangulation, which reflect its extrinsic curvature within the ambient 3d space, while the modulus is given by the angles within the 2d triangles, thus encoding the intrinsic geometry of the triangulation. Our formula can not cover the whole set of Ising zeroes, but we conjecture that a suitable complexification of these "real" zeroes would provide a more thorough formula. Nevertheless, in the thermodynamic limit, in the case of flat planar 2d triangulations, our Ising zeroes formula gives the critical couplings for isoradial graphs, confirming its generality. This approach shows an intricate, but precise, new relation between statistical mechanics and quantum geometry.

Surface diffusion and surface electromigration may lead to a morphological instability of thin solid films and nanowires. In this paper two nonlinear analyzes of a morphological instability are developed for a single-crystal cylindrical nanowire that is subjected to the axial current. These treatments extend the conventional linear stability analyzes without surface electromigration, that manifest a Rayleigh-Plateau instability. A weakly nonlinear analysis is done slightly above the Rayleigh-Plateau (longwave) instability threshold. It results in a one-dimensional Sivashinsky amplitude equation that describes a blow-up of a surface perturbation amplitude in a finite time. This is a signature of a formation of an axisymmetric spike singularity of a cylinder radius, which leads to a wire pinch-off and separation into a disjoint segments. The scaling analysis of the amplitude spike singularity is performed, and the time-and-electric field-dependent dimensions of the spike are characterized. A weakly nonlinear multi-scale analysis is done at the arbitrary distance above a longwave or a shortwave instability threshold. The time-and-electric field-dependent Fourier amplitudes of the major instability modes are derived and characterized.

It is commonly believed that logical states of quantum error-correcting codes have to be highly entangled such that codes capable of correcting more errors require more entanglement to encode a qubit. Here we show that this belief may or may not be true depending on a particular code. To this end, we characterize a tradeoff between the code distance $d$ quantifying the number of correctable errors, and geometric entanglement of logical states quantifying their maximal overlap with product states or more general "topologically trivial" states. The maximum overlap is shown to be exponentially small in $d$ for three families of codes: (1) low-density parity check (LDPC) codes with commuting check operators, (2) stabilizer codes, and (3) codes with a constant encoding rate. Equivalently, the geometric entanglement of any logical state of these codes grows at least linearly with $d$. On the opposite side, we also show that this distance-entanglement tradeoff does not hold in general. For any constant $d$ and $k$ (number of logical qubits), we show there exists a family of codes such that the geometric entanglement of some logical states approaches zero in the limit of large code length.

A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schroedinger equation, respectively, where the Schroedinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, Under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schroedinger equations that are a generalization of the Hartree-Fock equations are also obtained. These equations contain a quantum Coulomb's law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.

We expound upon our (polarization-free) definition of the quantization map in geometric quantization, which is justified using the Poisson sigma model and pieces together most known quantization schemes. We use it to obtain the noncommutative torus and a finite dimensional irreducible representation. We discuss invariance of polarization using Schur's lemma.

Given a quantum channel and a state which satisfy a fixed point equation approximately (say, up to an error $\varepsilon$), can one find a new channel and a state, which are respectively close to the original ones, such that they satisfy an exact fixed point equation? It is interesting to ask this question for different choices of constraints on the structures of the original channel and state, and requiring that these are also satisfied by the new channel and state. We affirmatively answer the above question, under fairly general assumptions on these structures, through a compactness argument. Additionally, for channels and states satisfying certain specific structures, we find explicit upper bounds on the distances between the pairs of channels (and states) in question. When these distances decay quickly (in a particular, desirable manner) as $\varepsilon\to 0$, we say that the original approximate fixed point equation is rapidly fixable. We establish rapid fixability, not only for general quantum channels, but also when the original and new channels are both required to be unitary, mixed unitary or unital. In contrast, for the case of bipartite quantum systems with channels acting trivially on one subsystem, we prove that approximate fixed point equations are not rapidly fixable. In this case, the distance to the closest channel (and state) which satisfy an exact fixed point equation can depend on the dimension of the quantum system in an undesirable way. We apply our results on approximate fixed point equations to the question of robustness of quantum Markov chains (QMC) and establish the following: For any tripartite quantum state, there exists a dimension-dependent upper bound on its distance to the set of QMCs, which decays to zero as the conditional mutual information of the state vanishes.