A density functional perspective for one-particle systems

Density functional theory is discussed in the context of one-particle systems. We show that the ground state density $\rho_0(x)$ and energy $E_0$ are simply related to a family of external potential energy functions with ground state wave functions $\psi_n(x) \propto \rho_0(x)^n$ and energies $E_n=2nE_0$ for certain integer values of $n$.Comment: 7 pages, ReVTeX4, submitted to Am. J. Phy

PENYELESAIAN NUMERIK PERSAMAAN DIFFERENSIAL BIASA MENGGUNAKAN JARINGAN FUNGSI RADIAL BASIS

Abstrak Di dalam artikel ini akan dijelaskan sebuah pendekatan numerik untuk penyelesaian persamaan differensial biasa (PDB) yang di dasarkan pada pendekatan suatu fungsi dan turunannya dengan menggunakan jaringan fungsi radial basis. Solusi dari persamaan tersebut, diperoleh dengan cara mengganti fungsi dan fungsi turunannya dengan sebuah fungsi pendekatan menggunakan jaringan fungsi radial basis (radial basis function). Hasil  yang diperoleh dengan menggunakan metode yang diusulkan ini,lebih baikkualitasnya jika dibandingkan dengan solusi yang diperoleh dengan menggunakan metode Runge-Kutta orde-4. Kelebihan dari metode ini adalah setiap fungsi dan turunannya dapat di dekati secara langsung dengan sebuah fungsi basis, sehingga untuk memperoleh solusi tidak diperlukan nilai awal.Hal ini selangkah lebih maju jika dibandingkan dengan metode-metode konvensional yang selalu memerlukan nilai awal. Disamping itu, jumlah komputasi yang di perlukan juga jauh lebih sedikit jika dibandingkan dengan jumlah komputasi pada metode-metode konvensional. Kata kunci : Penyelesaian Numerik, Persamaan Differensial, Jaringan  Syaraf Tiruan, Fungsi Radial Basis, RBF.   Abstract In this article will describe a numerical approach to the completion of ordinary differential equations (PDB), which is based on the approach of a function and its derivatives using radial basis function network. The solution of the equation, is obtained by replacing the function and its derivative function with a function approach uses radial basis function network (radial basis function). The results obtained using the proposed method, the more better quality when compared to solutions obtained using the Runge-Kutta method of order-4. The advantage of this method is that every function and its derivatives can be approached directly by a function of the base, so it as to obtain the initial value of the solution is not required. It is a step forward when compared to conventional methods always require an initial value. In addition, the amount of computing that need is also far less if compared to the amount of computing on conventional methods. Keywords: Numerical Resolution, Differential Equations, Neural Network, Radial Basis Function, RBF. Â

Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds

Feng Qi, Da-Wei Niu, and Dongkyu Lim, Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds, Miskolc Mathematical Notes 20 (2019), no. 2, 1129--1137; available online at https://doi.org/10.18514/MMN.2019.2976.International audienceIn the paper, starting from the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds, by virtue of the Fa\`a di Bruno formula, with the help of two identities for the Bell polynomials of the second kind, and making use of a new inversion theorem for combinatorial coefficients, the authors derive two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials of the first and second kinds

Vibrating quantum billiards on Riemannian manifolds

Quantum billiards provide an excellent forum for the analysis of quantum chaos. Toward this end, we consider quantum billiards with time-varying surfaces, which provide an important example of quantum chaos that does not require the semiclassical ($\hbar \longrightarrow 0$) or high quantum-number limits. We analyze vibrating quantum billiards using the framework of Riemannian geometry. First, we derive a theorem detailing necessary conditions for the existence of chaos in vibrating quantum billiards on Riemannian manifolds. Numerical observations suggest that these conditions are also sufficient. We prove the aforementioned theorem in full generality for one degree-of-freedom boundary vibrations and briefly discuss a generalization to billiards with two or more degrees-of-vibrations. The requisite conditions are direct consequences of the separability of the Helmholtz equation in a given orthogonal coordinate frame, and they arise from orthogonality relations satisfied by solutions of the Helmholtz equation. We then state and prove a second theorem that provides a general form for the coupled ordinary differential equations that describe quantum billiards with one degree-of-vibration boundaries. This set of equations may be used to illustrate KAM theory and also provides a simple example of semiquantum chaos. Moreover, vibrating quantum billiards may be used as models for quantum-well nanostructures, so this study has both theoretical and practical applications.Comment: 23 pages, 6 figures, a few typos corrected. To appear in International Journal of Bifurcation and Chaos (9/01