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Journal articles on the topic 'Polynomial Hamiltonians'

Author: Grafiati

Published: 14 March 2023

Last updated: 31 July 2025

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1

Marti, Lucas, Refik Mansuroglu, and Michael J. Hartmann. "Efficient Quantum Cooling Algorithm for Fermionic Systems." Quantum 9 (February 18, 2025): 1635. https://doi.org/10.22331/q-2025-02-18-1635.

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We present a cooling algorithm for ground state preparation of fermionic Hamiltonians. Our algorithm makes use of the Hamiltonian simulation of the considered system coupled to an ancillary fridge, which is regularly reset to its known ground state. We derive suitable interaction Hamiltonians that originate from ladder operators of the free theory and initiate resonant gaps between system and fridge. We further propose a spectroscopic scan to find the relevant eigenenergies of the system using energy measurements on the fridge. With these insights, we design a ground state cooling algorithm fo

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2

SILVER, R. N., and H. RÖDER. "DENSITIES OF STATES OF MEGA-DIMENSIONAL HAMILTONIAN MATRICES." International Journal of Modern Physics C 05, no. 04 (1994): 735–53. http://dx.doi.org/10.1142/s0129183194000842.

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We propose a statistical method to estimate densities of states (DOS) and thermodynamic functions of very large Hamiltonian matrices. Orthogonal polynomials are defined on the interval between lower and upper energy bounds. The DOS is represented by a kernel polynomial constructed out of polynomial moments of the DOS and modified to damp the Gibbs phenomenon. The moments are stochastically evaluated using matrixvector multiplications on Gaussian random vectors and the polynomial recurrence relations. The resulting kernel estimate is a controlled approximation to the true DOS, because it also p

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3

RÜHL, WERNER, and ALEXANDER TURBINER. "EXACT SOLVABILITY OF THE CALOGERO AND SUTHERLAND MODELS." Modern Physics Letters A 10, no. 29 (1995): 2213–21. http://dx.doi.org/10.1142/s0217732395002374.

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Translationally invariant symmetric polynomials as coordinates for N-body problems with identical particles are proposed. It is shown that in those coordinates the Calogero and Sutherland N-body Hamiltonians, after appropriate gauge transformations, can be presented as a quadratic polynomial in the generators of the algebra sl N in finitedimensional degenerate representation. The exact solvability of these models follows from the existence of the infinite flag of such representation spaces, preserved by the above Hamiltonians. A connection with Jack polynomials is discussed.

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4

Gosset, David, Jenish C. Mehta, and Thomas Vidick. "QCMA hardness of ground space connectivity for commuting Hamiltonians." Quantum 1 (July 14, 2017): 16. http://dx.doi.org/10.22331/q-2017-07-14-16.

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In this work we consider the ground space connectivity problem for commuting local Hamiltonians. The ground space connectivity problem asks whether it is possible to go from one (efficiently preparable) state to another by applying a polynomial length sequence of 2-qubit unitaries while remaining at all times in a state with low energy for a given HamiltonianH. It was shown in [Gharibian and Sikora, ICALP15] that this problem is QCMA-complete for general local Hamiltonians, where QCMA is defined as QMA with a classical witness and BQP verifier. Here we show that the commuting version of the pr

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5

Lu, Kang. "Completeness of Bethe Ansatz for Gaudin Models with gl(1|1) Symmetry and Diagonal Twists." Symmetry 15, no. 1 (2022): 9. http://dx.doi.org/10.3390/sym15010009.

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We studied the Gaudin models with gl(1|1) symmetry that are twisted by a diagonal matrix and defined on tensor products of polynomial evaluation gl(1|1)[t]-modules. Namely, we gave an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation gl(1|1)[t]-modules and showed that a bijection exists between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that eac

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6

Sokolov, A. V. "Polynomial supersymmetry for matrix Hamiltonians." Physics Letters A 377, no. 9 (2013): 655–62. http://dx.doi.org/10.1016/j.physleta.2013.01.012.

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7

Aharonov, Dorit, Michael Ben-Or, Fernando G. S. L. Brandão, and Or Sattath. "The Pursuit of Uniqueness: Extending Valiant-Vazirani Theorem to the Probabilistic and Quantum Settings." Quantum 6 (March 17, 2022): 668. http://dx.doi.org/10.22331/q-2022-03-17-668.

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Valiant-Vazirani showed in 1985 \cite{VV85} that solving NP with the promise that "yes" instances have only one witness is powerful enough to solve the entire NP class (under randomized reductions).We are interested in extending this result to the quantum setting. We prove extensions to the classes Merlin-Arthur MA and Quantum-Classical-Merlin-Arthur QCMA \cite{AN02}. Our results have implications for the complexity of approximating the ground state energy of a quantum local Hamiltonian with a unique ground state and an inverse polynomial spectral gap. We show that the estimation (to within po

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8

Bravyi, S., D. P. DiVincenzo, R. Oliveira, and B. M. Terhal. "The complexity of stoquastic local Hamiltonian problems." Quantum Information and Computation 8, no. 5 (2008): 361–85. http://dx.doi.org/10.26421/qic8.5-1.

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We study the complexity of the Local Hamiltonian Problem (denoted as LH-MIN) in the special case when a Hamiltonian obeys the condition that all off-diagonal matrix elements in the standard basis are real and non-positive. We will call such Hamiltonians, which are common in the natural world, stoquastic. An equivalent characterization of stoquastic Hamiltonians is that they have an entry-wise non-negative Gibbs density matrix for any temperature. We prove that LH-MIN for stoquastic Hamiltonians belongs to the complexity class \AM{}--- a probabilistic version of \NP{} with two rounds of communi

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9

UENO, YUICHI. "POLYNOMIAL HAMILTONIANS FOR QUANTUM PAINLEVÉ EQUATIONS." International Journal of Mathematics 20, no. 11 (2009): 1335–45. http://dx.doi.org/10.1142/s0129167x09005789.

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Recently, a quantum version of Painlevé equations from the point of view of their symmetries was proposed by Nagoya. These quantum Painlevé equations can be written as Hamiltonian systems with a (noncommutative) polynomial Hamiltonian H J . We give a characterization of the quantum Painlevé equations by certain holomorphic properties. Namely, we introduce canonical transformations such that the Painlevé Hamiltonian system is again transformed into a polynomial Hamiltonian system, and we show that the Hamiltonian can be uniquely characterized through this holomorphic property.

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10

Hussin, V., I. Marquette, and K. Zelaya. "Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials." Journal of Physics A: Mathematical and Theoretical 55, no. 4 (2022): 045205. http://dx.doi.org/10.1088/1751-8121/ac43cc.

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Abstract We extend and generalize the construction of Sturm–Liouville problems for a family of Hamiltonians constrained to fulfill a third-order shape-invariance condition and focusing on the ‘−2x/3’ hierarchy of solutions to the fourth Painlevé transcendent. Such a construction has been previously addressed in the literature for some particular cases but we realize it here in the most general case. The corresponding potential in the Hamiltonian operator is a rationally extended oscillator defined in terms of the conventional Okamoto polynomials, from which we identify three different zero-mod

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11

Ayryan, Edik, Michal Hnatic, Juha Honkonen, and Victor Malyutin. "Approximate Calculation of Functional Integrals Generated by Nonrelativistic and Relativistic Hamiltonians." Symmetry 15, no. 9 (2023): 1785. http://dx.doi.org/10.3390/sym15091785.

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The discussion revolves around the most recent outcomes in the realm of approximating functional integrals through calculations. Review of works devoted to the application of functional integrals in quantum mechanics and quantum field theory, nuclear physics and in other areas is presented. Methods obtained by the authors for approximate calculation of functional integrals generated by nonrelativistic Hamiltonians are given. One of the methods is based on the expansion in eigenfunctions of the Hamiltonian. In an alternate approach, the functional integrals are tackled using the semiclassical a

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12

GÉRARD, C., and A. PANATI. "SPECTRAL AND SCATTERING THEORY FOR SOME ABSTRACT QFT HAMILTONIANS." Reviews in Mathematical Physics 21, no. 03 (2009): 373–437. http://dx.doi.org/10.1142/s0129055x09003645.

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We introduce an abstract class of bosonic QFT Hamiltonians and study their spectral and scattering theories. These Hamiltonians are of the form H = dΓ(ω) + V acting on the bosonic Fock space Γ(𝔥), where ω is a massive one-particle Hamiltonian acting on 𝔥 and V is a Wick polynomial Wick(w) for a kernel w satisfying some decay properties at infinity. We describe the essential spectrum of H, prove a Mourre estimate outside a set of thresholds and prove the existence of asymptotic fields. Our main result is the asymptotic completeness of the scattering theory, which means that the CCR representati

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13

Vigo-Aguiar, M. I., M. E. Sansaturio, and J. M. Ferrándiz. "Integrability of Hamiltonians with polynomial potentials." Journal of Computational and Applied Mathematics 158, no. 1 (2003): 213–24. http://dx.doi.org/10.1016/s0377-0427(03)00467-9.

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14

Mingalev, Oleg V., Yurii N. Orlov, and Victor V. Vedenyapin. "Conservation laws for polynomial quantum Hamiltonians." Physics Letters A 223, no. 4 (1996): 246–50. http://dx.doi.org/10.1016/s0375-9601(96)00680-9.

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15

Palacián, Jesús, and Patricia Yanguas. "Equivariant N-Dof Hamiltonians Via Generalized Normal Forms." Communications in Contemporary Mathematics 05, no. 03 (2003): 449–80. http://dx.doi.org/10.1142/s0219199703001026.

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In the present paper we study polynomial Hamiltonian systems depending on one or various real parameters. We determine the values that these parameters should take in order to be able to construct formal (asymptotic) integrals of the system. In this respect, a method to calculate the formal integrals of a polynomial Hamiltonian vector field is presented. The original Hamilton function represents a family of dynamical systems composed by a principal part (quadratic terms) plus the perturbation (terms of degree three or bigger). We extend an integral of the principal part to the perturbed system

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16

Hall, Laurence S. "Invariants Polynomial in Momenta for Integrable Hamiltonians." Physical Review Letters 54, no. 7 (1985): 614–15. http://dx.doi.org/10.1103/physrevlett.54.614.

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17

BORESKOV, KONSTANTIN G., JUAN CARLOS LOPEZ VIEYRA, and ALEXANDER V. TURBINER. "SOLVABILITY OF THE F4 INTEGRABLE SYSTEM." International Journal of Modern Physics A 16, no. 29 (2001): 4769–801. http://dx.doi.org/10.1142/s0217751x0100550x.

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It is shown that the F4 rational and trigonometric integrable systems are exactly-solvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions are by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are invariant with respect to the Weyl group of F4 root system and can be obtained by averaging over an orbit of the Weyl group. An alternative way of finding these variables exploiting a property of duality of the F4 model is presented

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18

Bravyi, Sergey. "Monte Carlo simulation of stoquastic Hamiltonians." Quantum Information and Computation 15, no. 13&14 (2015): 1122–40. http://dx.doi.org/10.26421/qic15.13-14-3.

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Stoquastic Hamiltonians are characterized by the property that their off-diagonal matrix elements in the standard product basis are real and non-positive. Many interesting quantum models fall into this class including the Transverse field Ising Model (TIM), the Heisenberg model on bipartite graphs, and the bosonic Hubbard model. Here we consider the problem of estimating the ground state energy of a local stoquastic Hamiltonian $H$ with a promise that the ground state of $H$ has a non-negligible correlation with some `guiding' state that admits a concise classical description. A formalized ver

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19

Matushko, M. G., and V. V. Sokolov. "Polynomial forms for quantum elliptic Calogero–Moser Hamiltonians." Theoretical and Mathematical Physics 191, no. 1 (2017): 480–90. http://dx.doi.org/10.1134/s004057791704002x.

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20

Cao, Yudong, and Daniel Nagaj. "Perturbative gadgets without strong interactions." Quantum Information and Computation 15, no. 13&14 (2015): 1197–222. http://dx.doi.org/10.26421/qic15.13-14-7.

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Perturbative gadgets are used to construct a quantum Hamiltonian whose low-energy subspace approximates a given quantum $k$-local Hamiltonian up to an absolute error $\epsilon$. Typically, gadget constructions involve terms with large interaction strengths of order $\text{poly}(\epsilon^{-1})$. Here we present a 2-body gadget construction and prove that it approximates a Hamiltonian of interaction strength $\gamma = O(1)$ up to absolute error $\epsilon\ll\gamma$ using interactions of strength $O(\epsilon)$ instead of the usual inverse polynomial in $\epsilon$. A key component in our proof is a

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21

Gu, Shouzhen, Rolando D. Somma, and Burak Şahinoğlu. "Fast-forwarding quantum evolution." Quantum 5 (November 15, 2021): 577. http://dx.doi.org/10.22331/q-2021-11-15-577.

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We investigate the problem of fast-forwarding quantum evolution, whereby the dynamics of certain quantum systems can be simulated with gate complexity that is sublinear in the evolution time. We provide a definition of fast-forwarding that considers the model of quantum computation, the Hamiltonians that induce the evolution, and the properties of the initial states. Our definition accounts for any asymptotic complexity improvement of the general case and we use it to demonstrate fast-forwarding in several quantum systems. In particular, we show that some local spin systems whose Hamiltonians

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22

Palacián, Jesús, and Patricia Yanguas. "Reduction of Polynomial Planar Hamiltonians with Quadratic Unperturbed Part." SIAM Review 42, no. 4 (2000): 671–91. http://dx.doi.org/10.1137/s0036144599362327.

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23

Shi, Jicong, and Yiton T. Yan. "Explicitly integrable polynomial Hamiltonians and evaluation of Lie transformations." Physical Review E 48, no. 5 (1993): 3943–51. http://dx.doi.org/10.1103/physreve.48.3943.

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24

Kelbert, E., A. Hyder, F. Demir, Z. T. Hlousek, and Z. Papp. "Green's operator for Hamiltonians with Coulomb plus polynomial potentials." Journal of Physics A: Mathematical and Theoretical 40, no. 27 (2007): 7721–28. http://dx.doi.org/10.1088/1751-8113/40/27/020.

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25

Ukolov, Yu A., N. A. Chekanov, A. A. Gusev, V. A. Rostovtsev, S. I. Vinitsky, and Y. Uwano. "A REDUCE program for the normalization of polynomial Hamiltonians." Computer Physics Communications 166, no. 1 (2005): 66–80. http://dx.doi.org/10.1016/j.cpc.2004.10.010.

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26

Letourneau, P., and L. Vinet. "Superintegrable Systems: Polynomial Algebras and Quasi-Exactly Solvable Hamiltonians." Annals of Physics 243, no. 1 (1995): 144–68. http://dx.doi.org/10.1006/aphy.1995.1094.

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27

Maniraguha, Jean de Dieu, Krzysztof Marciniak, and Célestin Kurujyibwami. "Transforming Stäckel Hamiltonians of Benenti type to polynomial form." Advances in Theoretical and Mathematical Physics 26, no. 3 (2022): 711–34. http://dx.doi.org/10.4310/atmp.2022.v26.n3.a5.

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28

Cervia, Michael J., Amol V. Patwardhan, and A. B. Balantekin. "Symmetries of Hamiltonians describing systems with arbitrary spins." International Journal of Modern Physics E 28, no. 05 (2019): 1950032. http://dx.doi.org/10.1142/s0218301319500320.

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We consider systems where dynamical variables are the generators of the SU(2) group. A subset of these Hamiltonians is exactly solvable using the Bethe ansatz techniques. We show that Bethe ansatz equations are equivalent to polynomial relationships between the operator invariants, or equivalently, between eigenvalues of those invariants.

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29

Mostafazadeh, Ali. "Parasupersymmetric Quantum Mechanics and Indices of Fredholm Operators." International Journal of Modern Physics A 12, no. 15 (1997): 2725–39. http://dx.doi.org/10.1142/s0217751x9700150x.

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The general features of the degeneracy structure of (p = 2) parasupersymmetric quantum mechanics are employed to yield a classification scheme for the form of the parasupersymmetric Hamiltonians. The method is applied to parasupersymmetric systems whose Hamiltonian is the square root of a fourth order polynomial in the generators of the parasupersymmetry. These systems are interesting to study for they lead to the introduction of a set of topological invariants very similar to the Witten indices of ordinary supersymmetric quantum mechanics. The topological invariants associated with parasupers

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30

Anshu, Anurag, and Tony Metger. "Concentration bounds for quantum states and limitations on the QAOA from polynomial approximations." Quantum 7 (May 11, 2023): 999. http://dx.doi.org/10.22331/q-2023-05-11-999.

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We prove concentration bounds for the following classes of quantum states: (i) output states of shallow quantum circuits, answering an open question from \cite{DMRF22}; (ii) injective matrix product states; (iii) output states of dense Hamiltonian evolution, i.e. states of the form e&#x03B9;H(p)&#x22EF;e&#x03B9;H(1)|&#x03C8;0&#x27E9; for any n-qubit product state |&#x03C8;0&#x27E9;, where each H(i) can be any local commuting Hamiltonian satisfying a norm constraint, including dense Hamiltonians with interactions between any qubits. Our proofs use polynomial approxim

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31

Mastroianni, Rita, and Christos Efthymiopoulos. "Kolmogorov algorithm for isochronous Hamiltonian systems." Mathematics in Engineering 5, no. 2 (2022): 1–35. http://dx.doi.org/10.3934/mine.2023035.

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<abstract><p>We present a Kolmogorov-like algorithm for the computation of a normal form in the neighborhood of an invariant torus in 'isochronous' Hamiltonian systems, i.e., systems with Hamiltonians of the form $ {\mathcal{H}} = {\mathcal{H}}_0+\varepsilon {\mathcal{H}}_1 $ where $ {\mathcal{H}}_0 $ is the Hamiltonian of $ N $ linear oscillators, and $ {\mathcal{H}}_1 $ is expandable as a polynomial series in the oscillators' canonical variables. This method can be regarded as a normal form analogue of a corresponding Lindstedt method for coupled oscillators. We comment on the po

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32

Palacián, Jesús, and Patricia Yanguas. "Reduction of polynomial Hamiltonians by the construction of formal integrals." Nonlinearity 13, no. 4 (2000): 1021–54. http://dx.doi.org/10.1088/0951-7715/13/4/303.

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33

Rivera, A. L., N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf. "Evolution under polynomial Hamiltonians in quantum and optical phase spaces." Physical Review A 55, no. 2 (1997): 876–89. http://dx.doi.org/10.1103/physreva.55.876.

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34

Ramani, A., B. Dorizzi, B. Grammaticos, and J. Hietarinta. "Linearization on a submanifold of integrable Hamiltonians with polynomial potentials." Physica D: Nonlinear Phenomena 18, no. 1-3 (1986): 171–79. http://dx.doi.org/10.1016/0167-2789(86)90174-0.

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35

Basios, V., N. A. Chekanov, B. L. Markovski, V. A. Rostovtsev, and S. I. Vinitsky. "GITA: A REDUCE program for the normalization of polynomial Hamiltonians." Computer Physics Communications 90, no. 2-3 (1995): 355–68. http://dx.doi.org/10.1016/0010-4655(95)00080-y.

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36

Gharibian, Sevag, and Justin Yirka. "The complexity of simulating local measurements on quantum systems." Quantum 3 (September 30, 2019): 189. http://dx.doi.org/10.22331/q-2019-09-30-189.

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An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, [Ambainis, CCC 2014] defined the complexity class PQMA[log], and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian is PQMA[log]-complete. In this paper, we continue the study of PQMA[log], obtaining the following lower and upper bounds.Lower bounds (hardness results): - The PQMA[log]-completeness result of [Ambainis, CCC 2014] requires O(log⁡n)-local observables

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37

Ivanyos, G., A. B. Nagy, and L. Ronyai. "Constructions for quantum computing with symmetrized gates." Quantum Information and Computation 8, no. 5 (2008): 411–29. http://dx.doi.org/10.26421/qic8.5-4.

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We investigate constructions for simulating quantum computers with a polynomial slowdown on ensembles composed of qubits on which symmetrized versions of one- and two-qubit gates can be performed. The simulation is based on taking Lie commutators of symmetrized Hamiltonians to extract Hamiltonians at desired local positions. During the simulation, only a part of the qubits can be used for storing information, the others are left unchanged by the commutators. We propose constructions for various symmetry groups where a pretty large fraction of the qubits can be used. As a few of the other qubit

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38

Qi, Xiao-Liang, and Daniel Ranard. "Determining a local Hamiltonian from a single eigenstate." Quantum 3 (July 8, 2019): 159. http://dx.doi.org/10.22331/q-2019-07-08-159.

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We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is ``yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system lim

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DOLCINI, FABRIZIO, and ARIANNA MONTORSI. "INTEGRABLE EXTENDED HUBBARD HAMILTONIANS FROM SYMMETRIC GROUP EQUATIONS." International Journal of Modern Physics B 14, no. 17 (2000): 1719–28. http://dx.doi.org/10.1142/s0217979200001540.

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We consider the most general form of extended Hubbard Hamiltonian conserving the total spin and number of electrons, and find all the 1-dimensional completely integrable models which can be derived from first degree polynomial solution of the Yang–Baxter equation. It is shown that such models are 96. They are identified with the 16-dimensional representations of the class of solutions of symmetric group relations acting as generalized permutators. As particular examples, the EKS and some other known models are obtained. A method for determining the physical features of the above models is outl

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BRIHAYE, YVES. "QUASI-EXACTLY SOLVABLE MATRIX SCHRÖDINGER OPERATORS." Modern Physics Letters A 15, no. 26 (2000): 1647–53. http://dx.doi.org/10.1142/s0217732300002073.

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Two families of quasi-exactly solvable 2×2 matrix Schrödinger operators are constructed. The first one is based on a polynomial matrix potential and depends on three parameters. The second is a one-parameter generalization of the scalar Lamé equation. The relationship between these operators and QES Hamiltonians already considered in the literature is pointed out.

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41

LIN, SHAO-SHIUNG, and SHI-SHYR ROAN. "ALGEBRAIC GEOMETRY AND HOFSTADTER TYPE MODEL." International Journal of Modern Physics B 16, no. 14n15 (2002): 2097–106. http://dx.doi.org/10.1142/s0217979202011846.

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In this report, we study the algebraic geometry aspect of Hofstadter type models through the algebraic Bethe equation. In the diagonalization problem of certain Hofstadter type Hamiltonians, the Bethe equation is constructed by using the Baxter vectors on a high genus spectral curve. When the spectral variables lie on rational curves, we obtain the complete and explicit solutions of the polynomial Bethe equation; the relation with the Bethe ansatz of polynomial roots is discussed. Certain algebraic geometry properties of Bethe equation on the high genus algebraic curves are discussed in cooper

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42

Cariolaro, Gianfranco, and Alberto Vigato. "Evaluation of Hamiltonians from Complex Symplectic Matrices." Symmetry 15, no. 5 (2023): 1000. http://dx.doi.org/10.3390/sym15051000.

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Gaussian unitaries play a fundamental role in the field of continuous variables. In the general n mode, they may formulated by a second-order polynomial in the bosonic operators. Another important role related to Gaussian unitaries is played by the symplectic transformations in the phase space. The paper investigates the links between the two representations: the link from Hamiltonian to symplectic, governed by an exponential, and the link from symplectic to Hamiltonian, governed by a logarithm. Thus, an answer is given to the non-trivial question: which Hamiltonian produces a given symplectic

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43

Bibikov, Pavel Vitalievich. "On Classification of Polynomial Hamiltonians With Nondegenerate Linearly Stable Singular Point." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 1 (2019): 86–88. http://dx.doi.org/10.26907/0021-3446-2019-1-86-88.

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Bibikov, P. V. "On Classification of Polynomial Hamiltonians With Nondegenerate Linearly Stable Singular Point." Russian Mathematics 63, no. 1 (2019): 76–78. http://dx.doi.org/10.3103/s1066369x19010092.

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45

Leyvraz, F. "An approach for obtaining integrable Hamiltonians from Poisson-commuting polynomial families." Journal of Mathematical Physics 58, no. 7 (2017): 072902. http://dx.doi.org/10.1063/1.4996581.

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Rowe, D. J. "An algebraic approach to problems with polynomial Hamiltonians on Euclidean spaces." Journal of Physics A: Mathematical and General 38, no. 47 (2005): 10181–201. http://dx.doi.org/10.1088/0305-4470/38/47/009.

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47

Scarcella, Donato. "Asymptotically quasiperiodic solutions for time-dependent Hamiltonians." Nonlinearity 37, no. 6 (2024): 065005. http://dx.doi.org/10.1088/1361-6544/ad399f.

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Abstract Dynamical systems subject to perturbations that decay over time are relevant in describing many physical models, e.g. when considering the effect of a laser pulse on a molecule, in epidemiological studies, and celestial mechanics. For this purpose, we consider time-dependent Hamiltonian vector fields that are the sum of two components. The first has an invariant torus supporting quasiperiodic solutions, and the second decays as time tends to infinity. The time decay is modelled by functions satisfying suitable conditions verified by a proper polynomial decay in time. We prove the exis

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Giuliani, Clemens, Filippo Vicentini, Riccardo Rossi, and Giuseppe Carleo. "Learning ground states of gapped quantum Hamiltonians with Kernel Methods." Quantum 7 (August 29, 2023): 1096. http://dx.doi.org/10.22331/q-2023-08-29-1096.

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Neural network approaches to approximate the ground state of quantum hamiltonians require the numerical solution of a highly nonlinear optimization problem. We introduce a statistical learning approach that makes the optimization trivial by using kernel methods. Our scheme is an approximate realization of the power method, where supervised learning is used to learn the next step of the power iteration. We show that the ground state properties of arbitrary gapped quantum hamiltonians can be reached with polynomial resources under the assumption that the supervised learning is efficient. Using k

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Wahlberg, Patrik. "Propagation of polynomial phase space singularities for Schrödinger equations with quadratic Hamiltonians." MATHEMATICA SCANDINAVICA 122, no. 1 (2018): 107. http://dx.doi.org/10.7146/math.scand.a-97187.

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We study propagation of phase space singularities for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. Phase space singularities are measured by the lack of polynomial decay of given order in open cones in the phase space, which gives a parametrized refinement of the Gabor wave front set. The main result confirms the fundamental role of the singular space associated to the quadratic form for the propagation of phase space singularities. The singularities are contained in the singular space, and propagate in the intersectio

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50

Gusev, A. A., N. A. Chekanov, V. A. Rostovtsev, S. I. Vinitsky, and Y. Uwano. "A Comparison of Algorithms for the Normalization and Quantization of Polynomial Hamiltonians." Programming and Computer Software 30, no. 2 (2004): 75–82. http://dx.doi.org/10.1023/b:pacs.0000021264.38623.52.

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