We prove a Berry–Esseen theorem, a local central limit theorem and (local) large and (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain random (Young) tower extensions, which is the main focus of this paper. On the random towers we will obtain our

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves in the phase space

We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of \(\mathrm{GL}(n,\mathbb {C})\), which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard \(\mathrm{GL}(n

We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by 2n measurements up to a natural gauge. We also show that one can recover the full first-order term for a related equation having no gauge invariance, and that it

We study fluctuations of polynomial linear statistics for discrete Schrödinger operators with a random decaying potential. We describe a decomposition of the space of polynomials into a direct sum of three subspaces determining the growth rate of the variance of the corresponding linear statistic. In particular, each one of these subspaces defines a unique critical value for the decay-rate exponent

We prove existence and uniqueness of solutions of the semiclassical Einstein equation in flat cosmological spacetimes driven by a quantum massive scalar field with arbitrary coupling to the scalar curvature. In the semiclassical approximation, the backreaction of matter to curvature is taken into account by equating the Einstein tensor to the expectation values of the stress-energy tensor in a suitable

We construct fundamental solutions to the time-dependent Schrödinger equations on compact manifolds by the time-slicing approximation of the Feynman path integral. We show that the iteration of short-time approximate solutions converges to the fundamental solutions to the Schrödinger equations modified by the scalar curvature in the uniform operator topology from the Sobolev space to the space of square

Given an (irreducible) Möbius covariant net \({\mathcal {A}}\), we prove a Bisognano–Wichmann theorem for its categorical extension \({\mathscr {E}}^{\mathrm d}\) associated with the braided \(C^*\)-tensor category \(\mathrm {Rep}^{\mathrm d}({\mathcal {A}})\) of dualizable (more precisely, “dualized”) Möbius covariant \({\mathcal {A}}\)-modules. As a closely related result, we prove a (modified) Bisognano–Wichmann

We consider a one-dimensional classical many-body system with interaction potential of Lennard–Jones type in the thermodynamic limit at low temperature \(1/\beta \in (0,\infty )\). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains

We establish metric graph counterparts of Pleijel’s theorem on the asymptotics of the number of nodal domains \(\nu _n\) of the nth eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with \(L^1\)-potentials and a variety of vertex conditions as well as the p-Laplacian with natural vertex conditions, and without any assumptions on the lengths of

We consider separable 2D discrete Schrödinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies, we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J. Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to separable

We introduce and study two new classes of unital quantum channels. The first class describes a 2-parameter family of channels given by completely positive (CP) maps \(M_3({\mathbf {C}})\mapsto M_3({\mathbf {C}})\) which are both unital and trace-preserving. Almost every member of this family is factorizable and extreme in the set of CP maps which are both unital and trace-preserving, but is not extreme

We study an extension of the sandwiched Rényi relative entropies for normal positive functionals on a von Neumann algebra, for parameter values \(\alpha \in [1/2,1)\). This work is intended as a continuation of Jenčová (Ann Henri Poincaré 19:2513–2542, 2018), where the values \(\alpha >1\) were studied. We use the Araki–Masuda divergences of Berta et al. (Ann Henri Poincaré 9:1843–1867, 2018) and treat

We investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated

We study the spectral theory and the resolvent of the vector field generating the frame flow of closed hyperbolic three-dimensional manifolds on some family of anisotropic Sobolev spaces. We show the existence of a spectral gap and prove resolvent estimates using semiclassical methods.

The centralizer of the image of the diagonal embedding of \(U_q(\mathfrak {sl}_2)\) in the tensor product of three irreducible representations is examined in a Schur–Weyl duality spirit. The aim is to offer a description in terms of generators and relations. A conjecture in this respect is offered with the centralizers presented as quotients of the Askey–Wilson algebra. Support for the conjecture is

We prove the quantum Zeno effect in open quantum systems whose evolution, governed by quantum dynamical semigroups, is repeatedly and frequently interrupted by the action of a quantum operation. For the case of a quantum dynamical semigroup with a bounded generator, our analysis leads to a refinement of existing results and extends them to a larger class of quantum operations. We also prove the existence

In this paper, we compute the diffractive wave propagator of the Aharonov–Bohm effect (Aharonov and Bohm, Phys Rev 115(3):485, 1959) on \({\mathbf {R}}^2\) with a single solenoid using a technique of moving solenoid location. In addition, we compute the corresponding diffraction coefficient which is the principal symbol of the diffractive propagator. This paper proves the propagation of singularities

For unitary operators \(U_0,U\) in Hilbert spaces \({\mathcal {H}}_0,{\mathcal {H}}\) and identification operator \(J:{\mathcal {H}}_0\rightarrow {\mathcal {H}}\), we present results on the derivation of a Mourre estimate for U starting from a Mourre estimate for \(U_0\) and on the existence and completeness of the wave operators for the triple \((U,U_0,J)\). As an application, we determine spectral

We prove that the results in scattering theory that involve resonances are still valid for non-analytic potentials, even if the notion of resonance is not defined in this setting. More precisely, we show that if the potential of a semiclassical Schrödinger operator is supposed to be smooth and to decrease at infinity, the usual formulas relating scattering quantities and resonances still hold. The

We prove that for a sequence of nested sets \(\{U_n\}\) with \(\Lambda = \cap _n U_n\) a measure zero set, the localized escape rate converges to the extremal index of \(\Lambda \), provided that the dynamical system is \(\phi \)-mixing at polynomial speed. We also establish the general equivalence between the local escape rate for entry times and the local escape rate for returns. Examples include

A Correction to this paper has been published: https://doi.org/10.1007/s00023-020-01015-y

We develop a comprehensive framework in which the existence of solutions to the semiclassical Einstein equation (SCE) in cosmological spacetimes is shown. Different from previous work on this subject, we do not restrict to the conformally coupled scalar field and we admit the full renormalization freedom. Based on a regularization procedure, which utilizes homogeneous distributions and is equivalent

Through a finite-dimensional reduction of the difference Lamé equation, an elliptic analog of the Kac–Sylvester tridiagonal matrix is found. We solve the corresponding finite discrete Lamé equation by constructing an orthogonal basis of eigenvectors for this novel elliptic Kac–Sylvester matrix.

We consider the Schrödinger operator on the halfline with the potential \((m^2-\frac{1}{4})\frac{1}{x^2}\), often called the Bessel operator. We assume that m is complex. We study the domains of various closed homogeneous realizations of the Bessel operator. In particular, we prove that the domain of its minimal realization for \(|\mathrm{Re}(m)|<1\) and of its unique closed realization for \(\mathrm{Re}(m)>1\)

The Neumann points of an eigenfunction f on a quantum (metric) graph are the interior zeros of \(f'\). The Neumann domains of f are the sub-graphs bounded by the Neumann points. Neumann points and Neumann domains are the counterparts of the well-studied nodal points and nodal domains. We prove bounds on the number of Neumann points and properties of the probability distribution of this number. Two

We study bosonic quantum field theory on the double covering \(\widetilde{dS}_{2}\) of the two-dimensional de Sitter universe, identified to a coset space of the group \(\mathrm{SL}(2, {{\mathbb {R}}})\). The latter acts effectively on \(\widetilde{dS}_{2}\) and can be interpreted as it relativity group. The manifold is locally identical to the standard the Sitter spacetime \({dS}_2\); it is globally

We study monodromy reduction of Fuchsian connections from a sheave theoretic viewpoint, focusing on the case when a singularity of a special connection with four singularities has been resolved. The main tool of study is based on a bundle modification technique due to Drinfeld and Oblezin. This approach via invariant spaces and eigenvalue problems allows us not only to explain Erdélyi’s classical infinite

We provide an overview of the tools and techniques of resurgence theory used in the Borel-Écalle resummation method, which we then apply to the massless Wess–Zumino model. Starting from already known results on the anomalous dimension of the Wess–Zumino model, we solve its renormalisation group equation for the two-point function in a space of formal series. We show that this solution is 1-Gevrey and

We construct a one-parameter family of static and spherically symmetric solutions to the Einstein-Vlasov system bifurcating from the Schwarzschild spacetime. The constructed solutions have the property that the spatial support of the matter is a finite, spherically symmetric shell located away from the black hole. Our proof is mostly based on the analysis of the set of trapped timelike geodesics and

We study the semigroup generated by the hypoelliptic Laplacian on the circle and the maximal bounded holomorphic extension of this semigroup. Using an orthogonal decomposition into harmonic oscillators with complex shifts, we describe the domain of this extension and we show that boundedness in a half plane corresponds to absolute convergence of the expansion of the semigroup in eigenfunctions. This

A Correction to this paper has been published: 10.1007/s00023-020-01014-z

Lattices of polynomial KP and BKP \(\tau \)-functions labelled by partitions, with the flow variables equated to finite power sums, as well as associated multipair KP and multipoint BKP correlation functions, are expressed via generalizations of Jacobi’s bialternant formula for Schur functions and Nimmo’s Pfaffian ratio formula for Schur Q-functions. These are obtained by applying Wick’s theorem to

This paper revisits the proof of Anderson localization for multi-particle systems. We introduce a multi-particle version of the eigensystem multi-scale analysis by Elgart and Klein, which had previously been used for single-particle systems.

Random noncommutative geometry can be seen as a Euclidean path-integral quantization approach to the theory defined by the Spectral Action in noncommutative geometry (NCG). With the aim of investigating phase transitions in random NCG of arbitrary dimension, we study the nonperturbative Functional Renormalization Group for multimatrix models whose action consists of noncommutative polynomials in Hermitian

We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result

Wigner and Yanase conjectured the more general problem of the concavity of the function \(\rho \mapsto \text {Trace}\ \rho ^pK^*\rho ^{1-p}K\). Lieb gave an actual connection with something then known as the strong subadditivity of the quantum entropy conjecture, which was due to Ruelle and Robinson. Lieb proved the concavity of that function, and this deep result is known as the Lieb’s concavity theorem

It is emphasized that for interactions with derivative couplings, the Ward Identity (WI) securing the preservation of a global symmetry should be modified. Scalar QED is taken as an explicit example. More precisely, it is rigorously shown in scalar QED that the naive WI and the improved Ward Identity (‘Master Ward Identity’, MWI) are related to each other by a finite renormalization of the time-ordered

We consider a system of M particles in contact with a heat reservoir of \(N\gg M\) particles. The evolution in the system and the reservoir, together with their interaction, is modeled via the Kac’s master equation. We chose the initial distribution with total energy \(N+M\) and show that if the reservoir is initially in equilibrium, that is, if the initial distribution depends only on the energy of

We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates that are consistent with central limit theorems that have been established in the last years.

In Roesch (Proof of a null penrose conjecture using a new quasi-local mass, arXiv:1609.02875 [gr-qc], 2016), the notion of Double Convexity for a foliation of a conical null hypersurface was introduced to give a proof, if satisfied, of the Null Penrose Inequality. Double Convexity constrains the geometry of a Marginally Outer Trapped Surface (MOTS), called a quasi-round MOTS. In the first part of this

For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrödinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different

We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these

We consider interaction energies \(E_f[L]\) between a point \(O\in {\mathbb {R}}^d\), \(d\ge 2\), and a lattice L containing O, where the interaction potential f is assumed to be radially symmetric and decaying sufficiently fast at infinity. We investigate the conservation of optimality results for \(E_f\) when integer sublattices kL are removed (periodic arrays of vacancies) or substituted (periodic

The space of vacua of many four-dimensional, \({\mathcal {N}}=2\) supersymmetric gauge theories can famously be identified with a family of complex curves. For gauge group SU(2), this gives a fully explicit description of the low-energy effective theory in terms of an elliptic curve and associated modular fundamental domain. The two-dimensional space of vacua for gauge group SU(3) parametrizes an intricate

In Mizera (J High Energy Phys 8:097, 2017) , Sebastian Mizera discovered a tree expansion formula of a homology intersection number on the configuration space \(\mathcal {M}_{0,n}\). The formula originates in a study of Kawai–Lewellen–Tye relation in string theory. In this paper, we give an elementary proof of the formula. The basic ingredients are the combinatorics of the real moduli space \(\overline{\mathcal

We introduce a global thermostat on Kac’s 1D model for the velocities of particles in a space-homogeneous gas subjected to binary collisions, also interacting with a (local) Maxwellian thermostat. The global thermostat rescales the velocities of all the particles, thus restoring the total energy of the system, which leads to an additional drift term in the corresponding nonlinear kinetic equation.

We analyse the Casimir effect of two nonsingular centers of interaction in three space dimensions, using the framework developed by Herdegen. Our model is mathematically well-defined and all physical quantities are finite. We also consider a scaling limit, in which the problem tends to that with two Dirac \(\delta \)’s. In this limit the global Casimir energy diverges, but we obtain its asymptotic

We propose a new generalization to quantum states of the Wasserstein distance, which is a fundamental distance between probability distributions given by the minimization of a transport cost. Our proposal is the first where the transport plans between quantum states are in natural correspondence with quantum channels, such that the transport can be interpreted as a physical operation on the system

We investigate the spectrum of Schrödinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter tends to \(-\,\infty \), a narrow cluster of finitely many eigenvalues tends to \(-\,\infty \), while the eigenvalues above this cluster remain bounded from below. Certain “rogue” eigenvalues break away

We analyse the boundary structure of general relativity in the coframe formalism in the case of a lightlike boundary, i.e. when the restriction of the induced Lorentzian metric to the boundary is degenerate. We describe the associated reduced phase space in terms of constraints on the symplectic space of boundary fields. We explicitly compute the Poisson brackets of the constraints and identify the

In this note, we prove an existence result for the Einstein conformal constraint equations for metrics with vanishing Yamabe invariant assuming that the mean curvature satisfies an explicit near-CMC condition and that the TT-tensor is small in \(L^2\).

We consider independent electrons in a periodic crystal in their ground state, and turn on a uniform electric field at some prescribed time. We rigorously define the current per unit volume and study its properties using both linear response and adiabatic theory. Our results provide a unified framework for various phenomena such as the quantization of Hall conductivity of insulators with broken time-reversibility

We consider rapidly mixing dynamical systems and link the decay of the shortest distance between multiple orbits with the generalized fractal dimension. We apply this result to multidimensional expanding maps and extend it to the realm of random dynamical systems. For random sequences, we obtain a relation between the longest common substring between multiple sequences and the generalized Rényi entropy

We prove that the only asymptotically flat spacetimes with a suitably regular event horizon, in a generalised Majumdar–Papapetrou class of solutions to higher-dimensional Einstein–Maxwell theory, are the standard multi-black holes. The proof involves a careful analysis of the near-horizon geometry and an extension of the positive mass theorem to Riemannian manifolds with conical singularities. This

We define a random model for the moments of the new eigenfunctions of a point scatterer on a two-dimensional rectangular flat torus. In the deterministic setting, S̆eba conjectured these moments to be asymptotically Gaussian, in the semi-classical limit. This conjecture was disproved by Kurlberg–Ueberschär on Diophantine tori. In our model, we describe the accumulation points in distribution of the

Equation (5.20), should read as follows.

A class of deep Boltzmann machines is considered in the simplified framework of a quenched system with Gaussian noise and independent entries. The quenched pressure of a K-layers spin glass model is studied allowing interactions only among consecutive layers. A lower bound for the pressure is found in terms of a convex combination of K Sherrington–Kirkpatrick models and used to study the annealed and

We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we introduce the K-curvature and K-torsion of a Courant algebroid connection. These are