On spectral asymptotic of quasi-exactly solvable quartic potential

Motivated by the earlier results of Masoero and De Benedetti (Nonlinearity 23:2501, 2010) and Shapiro et al. (Commun Math Phys 311(2):277–300, 2012), we discuss below the asymptotic of the solvable part of the spectrum for the quasi-exactly solvable quartic oscillator. In particular, we formulate a conjecture on the coincidence of the asymptotic shape of the level crossings of the latter oscillator with the asymptotic shape of zeros of the Yablonskii–Vorob’ev polynomials. Further we present a numerical study of the spectral monodromy for the oscillator in question.

Eigenstate Thermalization Hypothesis for Wigner Matrices

We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with very high probability and with an optimal error inversely proportional to the square root of the dimension. Our theorem thus rigorously verifies the Eigenstate Thermalisation Hypothesis by Deutsch (Phys Rev A 43:2046–2049, 1991) for the simplest chaotic quantum system, the Wigner ensemble. In mathematical terms, we prove the strong form of Quantum Unique Ergodicity (QUE) with an optimal convergence rate for all eigenvectors simultaneously, generalizing previous probabilistic QUE results in Bourgade and Yau (Commun Math Phys 350:231–278, 2017) and Bourgade et al. (Commun Pure Appl Math 73:1526–1596, 2020).

Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation

The super-Macdonald polynomials, introduced by Sergeev and Veselov (Commun Math Phys 288: 653–675, 2009), generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed Macdonald–Ruijsenaars operators introduced by the same authors in Sergeev and Veselov (Commun Math Phys 245: 249–278, 2004). We introduce a Hermitian form on the algebra spanned by the super-Macdonald polynomials, prove their orthogonality, compute their (quadratic) norms explicitly, and establish a corresponding Hilbert space interpretation of the super-Macdonald polynomials and deformed Macdonald–Ruijsenaars operators. This allows for a quantum mechanical interpretation of the models defined by the deformed Macdonald–Ruijsenaars operators. Motivated by recent results in the nonrelativistic ($$q\rightarrow 1$$ q → 1 ) case, we propose that these models describe the particles and anti-particles of an underlying relativistic quantum field theory, thus providing a natural generalisation of the trigonometric Ruijsenaars model.

Fluctuations of extreme eigenvalues of sparse Erdős–Rényi graphs

We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős–Rényi graph $${{\mathcal {G}}}(N,p)$$ G ( N , p ) . We show that if $$N^{\varepsilon } \leqslant Np \leqslant N^{1/3-\varepsilon }$$ N ε ⩽ N p ⩽ N 1 / 3 - ε then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result (Huang et al. in Ann Prob 48:916–962, 2020) on the fluctuations of the extreme eigenvalues from $$Np \geqslant N^{2/9 + \varepsilon }$$ N p ⩾ N 2 / 9 + ε down to the optimal scale $$Np \geqslant N^{\varepsilon }$$ N p ⩾ N ε . The main technical achievement of our proof is a rigidity bound of accuracy $$N^{-1/2-\varepsilon } (Np)^{-1/2}$$ N - 1 / 2 - ε ( N p ) - 1 / 2 for the extreme eigenvalues, which avoids the $$(Np)^{-1}$$ ( N p ) - 1 -expansions from Erdős et al. (Ann Prob 41:2279–2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543–616, 2018). Our result is the last missing piece, added to Erdős et al. (Commun Math Phys 314:587–640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for $$Np \geqslant N^{\varepsilon }$$ N p ⩾ N ε .

Invariance Principle for the Random Lorentz Gas—Beyond the Boltzmann-Grad Limit

We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius r, placed according to a Poisson point process of density $$\varrho $$ ϱ , in the limit $$\varrho \rightarrow \infty $$ ϱ → ∞ , $$r\rightarrow 0$$ r → 0 , $$\varrho r^{2}\rightarrow 1$$ ϱ r 2 → 1 up to time scales of order $$T=o(r^{-2}\left| {\log r}\right| ^{-2})$$ T = o ( r - 2 log r - 2 ) . To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (Phys Rev 185:308–322, 1969, Nota Interna Univ di Roma 358, 1970, Statistical mechanics. A short treatise. Theoretical and mathematical physics series, Springer, Berlin, 1999), Spohn (Commun Math Phys 60:277–290, 1978, Rev Mod Phys 52:569–611, 1980) and Boldrighini–Bunimovich–Sinai (J Stat Phys 32:477–501, 1983). The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann-Grad) limit are taken simultaneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling. Similar results have been earlier obtained for the weak coupling limit of classical and quantum random Lorentz gas, by Komorowski–Ryzhik (Commun Math Phys 263:277–323, 2006), respectively, Erdős–Salmhofer–Yau (Acta Math 200:211–277, 2008, Commun Math Phys 271:1–53, 2007). However, the following are substantial differences between our work and these ones: (1) The physical setting is different: low density rather than weak coupling. (2) The method of approach is different: probabilistic coupling rather than analytic/perturbative. (3) Due to (2), the time scale of validity of our diffusive approximation—expressed in terms of the kinetic time scale—is much longer and fully explicit.

Best constants in Sobolev and Gagliardo–Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations

In this paper the dependence of the best constants in Sobolev and Gagliardo–Nirenberg inequalities on the precise form of the Sobolev space norm is investigated. The analysis is carried out on general graded Lie groups, thus including the cases of $$\mathbb {R}^n$$ R n , Heisenberg, and general stratified Lie groups, in all these cases the results being new. The Sobolev norms may be defined in terms of Rockland operators, i.e. the hypoelliptic homogeneous left-invariant differential operators on the group. The best constants are expressed in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The orders of these equations can be high depending on the Sobolev space order in the Sobolev or Gagliardo–Nirenberg inequalities, or may be fractional. Applications are obtained also to equations with lower order terms given by different hypoelliptic operators. Already in the case of $${\mathbb {R}}^n$$ R n , the obtained results extend the classical relations by Weinstein (Commun Math Phys 87(4):567–576 (1982/1983)) to a wide range of nonlinear elliptic equations of high orders with elliptic low order terms and a wide range of interpolation inequalities of Gagliardo–Nirenberg type. However, the proofs are different from those in Weinstein (Commun Math Phys 87(4):567–576 (1982/1983)) because of the impossibility of using the rearrangement inequalities already in the setting of the Heisenberg group. The considered class of graded groups is the most general class of nilpotent Lie groups where one can still consider hypoelliptic homogeneous invariant differential operators and the corresponding subelliptic differential equations.

Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

We introduce a space-inhomogeneous generalization of the dynamics on interlacing arrays considered by Borodin and Ferrari (Commun Math Phys 325:603–684, 2014). We show that for a certain class of initial conditions the point process associated with the dynamics has determinantal correlation functions, and we calculate explicitly, in the form of a double contour integral, the correlation kernel for one of the most classical initial conditions, the densely packed. En route to proving this, we obtain some results of independent interest on non-intersecting general pure-birth chains, that generalize the Charlier process, the discrete analogue of Dyson’s Brownian motion. Finally, these dynamics provide a coupling between the inhomogeneous versions of the TAZRP and PushTASEP particle systems which appear as projections on the left and right edges of the array, respectively.

Quasi-normal modes for Dirac fields in the Kerr–Newman–de Sitter black holes

We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of fixed width for Dirac fields in slowly rotating Kerr–Newman–de Sitter black holes. The resonances split in a way similar to the Zeeman effect. The method is based on the extension to Dirac operators of techniques applied by Dyatlov in [Quasi-normal modes and exponential energy decay for the Kerr–de Sitter black hole, Commun. Math. Phys. 306(1) (2011) 119–163; Asymptotic distribution of quasi-normal modes for Kerr–de Sitter black holes, Ann. Henri Poincaré 13(5) (2012) 1101–1166] to the (uncharged) Kerr–de Sitter black holes. We show that the mass of the Dirac field does not have an effect on the two leading terms in the expansions of resonances. We give an expansion of the solution of the evolution equation for the Dirac fields in the outer region of the slowly rotating Kerr–Newman–de Sitter black hole which implies the exponential decay of the local energy. Moreover, using the [Formula: see text]-normal hyperbolicity of the trapped set and applying the techniques from [Asymptotics of linear waves and resonances with applications to black holes, Commun. Math. Phys. 335 (2015) 1445–1485; Resonance projectors and asymptotics for [Formula: see text]-normally hyperbolic trapped sets, J. Amer. Math. Soc. 28 (2015) 311–381], we give location of the resonance free band and the Weyl-type formula for the resonances in the band near the real axis.

QCD breaks Lorentz invariance and colour

In the previous work [A. P. Balachandran and S. Vaidya, Eur. Phys. J. Plus 128, 118 (2013)], we have argued that the algebra of non-Abelian superselection rules is spontaneously broken to its maximal Abelian subalgebra, that is, the algebra generated by its completing commuting set (the two Casimirs, isospin and a basis of its Cartan subalgebra). In this paper, alternative arguments confirming these results are presented. In addition, Lorentz invariance is shown to be broken in quantum chromodynamics (QCD), just as it is in quantum electrodynamics (QED). The experimental consequences of these results include fuzzy mass and spin shells of coloured particles like quarks, and decay life times which depend on the frame of observation [D. Buchholz, Phys. Lett. B 174, 331 (1986); D. Buchholz and K. Fredenhagen, Commun. Math. Phys. 84, 1 (1982; J. Fröhlich, G. Morchio and F. Strocchi, Phys. Lett. B 89, 61 (1979); A. P. Balachandran, S. Kürkçüoğlu, A. R. de Queiroz and S. Vaidya, Eur. Phys. J. C 75, 89 (2015); A. P. Balachandran, S. Kürkçüoğlu and A. R. de Queiroz, Mod. Phys. Lett. A 28, 1350028 (2013)]. In a paper under preparation, these results are extended to the ADM Poincaré group and the local Lorentz group of frames. The renormalisation of the ADM energy by infrared gravitons is also studied and estimated.

On the Witten–Reshetikhin–Turaev invariants of torus bundles

By methods similar to those used by L. Jeffrey [L. C. Jeffrey, Chern–Simons–Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys.147 (1992) 563–604], we compute the quantum SU (N)-invariants for mapping tori of trace 2 homeomorphisms of a genus 1 surface when N = 2, 3 and discuss their asymptotics. In particular, we obtain directly a proof of a version of Witten's asymptotic expansion conjecture for these 3-manifolds. We further prove the growth rate conjecture for these 3-manifolds in the SU(2) case, where we also allow the 3-manifolds to contain certain knots. In this case we also discuss trace -2 homeomorphisms, obtaining — in combination with Jeffrey's results — a proof of the asymptotic expansion conjecture for all torus bundles.