Some specific solutions to the translation-invariant $N$-body harmonic oscillator Hamiltonian

• URL: http://arxiv.org/abs/2108.05171v1
• Date: Wed, 11 Aug 2021 11:58:05 GMT
• Title: Some specific solutions to the translation-invariant $N$-body harmonic oscillator Hamiltonian
• Authors: Cintia T. Willemyns and Claude Semay
• Abstract summary: We show that the diagonalization of a matrix $mathbbJ$ can be analytically solved. We present analytical expressions for the energies under those constraints.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The resolution of the Schr\"odinger equation for the translation-invariant $N$-body harmonic oscillator Hamiltonian in $D$ dimensions with one-body and two-body interactions is performed by diagonalizing a matrix $\mathbb{J}$ of order $N-1$. It has been previously established that the diagonalization can be analytically performed in specific situations, such as for $N \le 5$ or for $N$ identical particles. We show that the matrix $\mathbb{J}$ is diagonal, and thus the problem can be analytically solved, for any number of arbitrary masses provided some specific relations exist between the coupling constants and the masses. We present analytical expressions for the energies under those constraints.

Related papers

• Exact solution of the isotropic and anisotropic Hamiltonian of two coupled harmonic oscillators [0.0]
  We write its Hamiltonian in terms of the boson generators of the $SU(1,1)$ and $SU(2)$ groups. We use the one boson and two boson realizations of the $su (1,1)$ Lie algebra to apply three tilting transformations to diagonalize the original Hamiltonian. These transformations let us to obtain the exact solutions of the isotropic and the anisotropic cases.
  arXiv  Detail & Related papers  (2024-09-15T20:53:42Z)
• Remarks on effects of projective phase on eigenstate thermalization hypothesis [0.0]
  We consider $mathbbZ_NtimesmathbbZ_N$ symmetries with nontrivial projective phases. We also perform numerical analyses for $ (1+1)$-dimensional spin chains and the $ (2+1)$-dimensional lattice gauge theory.
  arXiv  Detail & Related papers  (2023-10-17T17:36:37Z)
• Non-standard quantum algebras and finite dimensional $\mathcal{PT}$-symmetric systems [0.0]
  We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard $U_z(sl(2, mathbb R))$ Hopf algebra deformation. We show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits.
  arXiv  Detail & Related papers  (2023-09-26T23:17:22Z)
• Rigorous derivation of the Efimov effect in a simple model [68.8204255655161]
  We consider a system of three identical bosons in $mathbbR3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$.
  arXiv  Detail & Related papers  (2023-06-21T10:11:28Z)
• Near-optimal fitting of ellipsoids to random points [68.12685213894112]
  A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
  arXiv  Detail & Related papers  (2022-08-19T18:00:34Z)
• Towards Antisymmetric Neural Ansatz Separation [48.80300074254758]
  We study separations between two fundamental models of antisymmetric functions, that is, functions $f$ of the form $f(x_sigma(1), ldots, x_sigma(N)) These arise in the context of quantum chemistry, and are the basic modeling tool for wavefunctions of Fermionic systems.
  arXiv  Detail & Related papers  (2022-08-05T16:35:24Z)
• Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions [77.34726150561087]
  We discuss some properties of a model Hamiltonian for a system of three bosons interacting via zero-range forces in three dimensions. In particular, starting from a suitable quadratic form $Q$, the self-adjoint and bounded from below Hamiltonian $mathcal H$ can be constructed. We show that the threshold value $gamma_c$ is optimal, in the sense that the quadratic form $Q$ is unbounded from below if $gammagamma_c$.
  arXiv  Detail & Related papers  (2022-07-01T10:01:14Z)
• $O(N^2)$ Universal Antisymmetry in Fermionic Neural Networks [107.86545461433616]
  We propose permutation-equivariant architectures, on which a determinant Slater is applied to induce antisymmetry. FermiNet is proved to have universal approximation capability with a single determinant, namely, it suffices to represent any antisymmetric function. We substitute the Slater with a pairwise antisymmetry construction, which is easy to implement and can reduce the computational cost to $O(N2)$.
  arXiv  Detail & Related papers  (2022-05-26T07:44:54Z)
• From quartic anharmonic oscillator to double well potential [77.34726150561087]
  It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction $Psi_ao(u)$, obtained recently, it is possible to get highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.
  arXiv  Detail & Related papers  (2021-10-30T20:16:27Z)
• Conformal generation of an exotic rotationally invariant harmonic oscillator [0.0]
  An exotic rotationally invariant harmonic oscillator (ERIHO) is constructed by applying a non-unitary isotropic conformal bridge transformation (CBT) to a free planar particle. We show that the ERIHO system is transformed by a peculiar unitary transformation into the anisotropic harmonic oscillator generated, in turn, by anisotropic CBT.
  arXiv  Detail & Related papers  (2021-03-13T17:09:43Z)
• Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors [0.0]
  We prove that in order that the description of the spin will be relativistic, it is not enough to preserve the projection lattice equipped with its natural partial order and denoteity. This, in particular, extends a result of Moln'ar to the wider setting of atomic JBW$*$-triples not containing rank-one Cartan factors.
  arXiv  Detail & Related papers  (2021-01-03T17:21:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.