New Concept of Solvability in Quantum Mechanics

In a pre-selected Hilbert space of quantum states the unitarity of the evolution is usually guaranteed via a pre-selection of the generator (i.e., of the Hamiltonian operator) in self-adjoint form. In fact, the simultaneous use of both of these pre-selections is overrestrictive. One should be allowed to make a given Hamiltonian self-adjoint only after an {\em ad hoc} generalization of Hermitian conjugation. We argue that in the generalized, hidden-Hermiticity scenario with nontrivial metric, the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of Hamiltonian) requires a generalization allowing for a non-numerical form of metric. A few illustrative solvable quantum models of this type are presented.Comment: 24 pages, 8 figure

Two-parameter Sturm-Liouville problems

This paper deals with the computation of the eigenvalues of two-parameter Sturm- Liouville (SL) problems using the Regularized Sampling Method, a method which has been effective in computing the eigenvalues of broad classes of SL problems (Singular, Non-Self-Adjoint, Non-Local, Impulsive,...). We have shown, in this work that it can tackle two-parameter SL problems with equal ease. An example was provided to illustrate the effectiveness of the method.Comment: 9 page

Fisher information and asymptotic normality in system identification for quantum Markov chains

This paper deals with the problem of estimating the coupling constant $\theta$ of a mixing quantum Markov chain. For a repeated measurement on the chain's output we show that the outcomes' time average has an asymptotically normal (Gaussian) distribution, and we give the explicit expressions of its mean and variance. In particular we obtain a simple estimator of $\theta$ whose classical Fisher information can be optimized over different choices of measured observables. We then show that the quantum state of the output together with the system, is itself asymptotically Gaussian and compute its quantum Fisher information which sets an absolute bound to the estimation error. The classical and quantum Fisher informations are compared in a simple example. In the vicinity of $\theta=0$ we find that the quantum Fisher information has a quadratic rather than linear scaling in output size, and asymptotically the Fisher information is localised in the system, while the output is independent of the parameter.Comment: 10 pages, 2 figures. final versio

Exact Solution of the Quantum Calogero-Gaudin System and of its q-Deformation

A complete set of commuting observables for the Calogero-Gaudin system is diagonalized, and the explicit form of the corresponding eigenvalues and eigenfunctions is derived. We use a purely algebraic procedure exploiting the co-algebra invariance of the model; with the proper technical modifications this procedure can be applied to the $q-$deformed version of the model, which is then also exactly solved.Comment: 20 pages Late

On uncertainty quantification of eigenvalues and eigenspaces with higher multiplicity

We consider generalized operator eigenvalue problems in variational form with random perturbations in the bilinear forms. This setting is motivated by variational forms of partial differential equations with random input data. The considered eigenpairs can be of higher but finite multiplicity. We investigate stochastic quantities of interest of the eigenpairs and discuss why, for multiplicity greater than 1, only the stochastic properties of the eigenspaces are meaningful, but not the ones of individual eigenpairs. To that end, we characterize the Fr\'echet derivatives of the eigenpairs with respect to the perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result, we prove local analyticity of the eigenspaces. Based on the Fr\'echet derivatives of the eigenpairs we discuss a meaningful Monte Carlo sampling strategy for multiple eigenvalues and develop an uncertainty quantification perturbation approach. Numerical examples are presented to illustrate the theoretical results

Model Order Reduction based on Proper Generalized Decomposition for the Propagation of Uncertainties in Structural Dynamics

International audienceA priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework. The Proper Generalized Decomposition method is used to construct a quasi-optimal separated representation of the random solution at some frequency samples. At each frequency, an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions. An extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure which appears to be accurate on the whole frequency band under study and for all values of input random parameters. This strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties. This reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest. The dynamic response being highly nonlinear with respect to the input random parameters, a second level of separation of variables is introduced for the representation of functions of multiple random parameters, thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension

Spectral tensor-train decomposition

The accurate approximation of high-dimensional functions is an essential task in uncertainty quantification and many other fields. We propose a new function approximation scheme based on a spectral extension of the tensor-train (TT) decomposition. We first define a functional version of the TT decomposition and analyze its properties. We obtain results on the convergence of the decomposition, revealing links between the regularity of the function, the dimension of the input space, and the TT ranks. We also show that the regularity of the target function is preserved by the univariate functions (i.e., the "cores") comprising the functional TT decomposition. This result motivates an approximation scheme employing polynomial approximations of the cores. For functions with appropriate regularity, the resulting \textit{spectral tensor-train decomposition} combines the favorable dimension-scaling of the TT decomposition with the spectral convergence rate of polynomial approximations, yielding efficient and accurate surrogates for high-dimensional functions. To construct these decompositions, we use the sampling algorithm \texttt{TT-DMRG-cross} to obtain the TT decomposition of tensors resulting from suitable discretizations of the target function. We assess the performance of the method on a range of numerical examples: a modifed set of Genz functions with dimension up to $100$, and functions with mixed Fourier modes or with local features. We observe significant improvements in performance over an anisotropic adaptive Smolyak approach. The method is also used to approximate the solution of an elliptic PDE with random input data. The open source software and examples presented in this work are available online.Comment: 33 pages, 19 figure

Combinatorics of Classical Unitary Invariant Ensembles and Integrable Systems

The first part of this thesis is devoted to the combinatorics, geometry, and effective computation of correlators of unitary invariant ensembles of random hermitian matrices with classical potentials. The main results are the subject of the publications [7, 8] with my supervisors T.~Grava and G.~Ruzza, and are summarized as follows. We provide generating functions for correlators of general Hermitian matrix models; formulae of this sort have already appeared in the literature [1, 5], we rederive them here with different methods which lend themselves to further generalizations. Such formulae are not recursive in the genus and hence particularly effective. Moreover, these formulae express the correlators of classical unitary ensembles as linear combinations of products of discrete hypergeometric polynomials; this generalizes relations to discrete orthogonal polynomials for the one-point correlators \langle \tr M^k \rangle of the classical ensembles recently discovered by Cunden et al. [3]. Hence, we turn our attention on the combinatorial interpretation of correlators for the Laguerre and Jacobi ensembles. We prove that the coefficients in the topological expansion of Jacobi correlators are multiparametric single Hurwitz numbers involving combinations of triple monotone Hurwitz numbers. Via a simple limit, this reproduces formulae of [2] on the Laguerre ensemble. This completes the combinatorial interpretation of correlators of unitary ensembles with classical potential. Combining results of Dubrovin et al. [4], and of Norbury [10] connecting integrable systems with enumerative geometry, we obtain ELSV-like formulae linking the multiparametric single Hurwitz numbers of LUE and JUE respectively to cubic Hodge integrals and $\Theta$-GW invariants. In the second part of the thesis we analyse various integrable dynamical systems from a probabilistic point of view. Specifically, we study the spectrum of their random Lax Matrix equipped with the associated Gibbs Measure, in the spirit of [9, 11]. This is the content of the preprint [6], in collaboration with T.~Grava, G.~Gubbiotti and G.~Mazzuca. We explicitly compute the density of states for the exponential Toda lattice and the Volterra lattice showing they are connected to the Laguerre $\beta$-ensemble at high temperatures and the $\beta$-antisymmetric Gaussian ensemble at high temperatures respectively. For generalizations of these system we derive numerically their density of states and compute their ground states. [1] M. Bertola, B. Dubrovin, and D. Yang, Correlation functions of the KdV hierarchy and applications to intersection numbers over Mg,n, Phys. D, 327 (2016), pp. 30–57. [2] F. D. Cunden, A. Dahlqvist, and N. O’Connell, Integer moments of complex Wishart matrices and Hurwitz numbers, Ann. Inst. Henri Poincar ́e D, 8 (2021), pp. 243–268. [3] F. D. Cunden, F. Mezzadri, N. O’Connell, and N. Simm, Moments of random matrices and hypergeometric orthogonal polynomials, Comm. Math. Phys, 369 (2019), pp. 1091–1145. [4] B. Dubrovin, S. Q. Liu, D. Yang, and Y. Zhang, Hodge-GUE correspondence and the discrete KdV equation, Comm. Math. Phys, 379 (2020), pp. 461–490. [5] B. Eynard, T. Kimura, and S. Ribault, Random matrices, arXiv preprint arXiv:1510.04430, (2015). [6] M. Gisonni, T. Grava, G. Gubbiotti, and G. Mazzuca, Discrete integrable systems and random Lax matrices, arXiv preprint arXiv:2206.15371, (2022). [7] M. Gisonni, T. Grava, and G. Ruzza, Laguerre ensemble: Correlators, Hurwitz numbers and Hodge integrals, Ann. Henri Poincar ́e, 21 (2020), pp. 3285–3339. [8] M. Gisonni, T. Grava, and G. Ruzza, Jacobi ensemble, Hurwitz numbers and Wilson polynomials, Lett. Math. Phys., 111 (2021), pp. 1–38. [9] T. Grava and G. Mazzuca, Generalized gibbs ensemble of the Ablowitz-Ladik lattice, Circular β-ensemble and double confluent Heun equation, arXiv preprint: 2107.02303, (2021). [10] P. Norbury, Gromov-Witten invariants of P1 coupled to a KdV tau function, Adv. Math., 399 (2022), p. 108227. [11] H. Spohn, Generalized Gibbs ensembles of the classical Toda chain, J. Stat. Phys., 180 (2020), pp. 4–2