98,041 research outputs found
Relativistic N-boson systems bound by pair potentials V(r_{ij}) = g(r_{ij}^2)
We study the lowest energy E of a relativistic system of N identical bosons
bound by pair potentials of the form V(r_{ij}) = g(r_{ij}^2) in three spatial
dimensions. In natural units hbar = c = 1 the system has the semirelativistic
`spinless-Salpeter' Hamiltonian H = \sum_{i=1}^N \sqrt{m^2 + p_i^2} +
\sum_{j>i=1}^N g(|r_i - r_j|^2), where g is monotone increasing and has
convexity g'' >= 0. We use `envelope theory' to derive formulas for general
lower energy bounds and we use a variational method to find complementary upper
bounds valid for all N >= 2. In particular, we determine the energy of the
N-body oscillator g(r^2) = c r^2 with error less than 0.15% for all m >= 0, N
>= 2, and c > 0.Comment: 15 pages, 4 figure
A chain theorem for 4-connected matroids
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On contracting hyperplane elements from a 3-connected matroid
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Energy bounds for the spinless Salpeter equation: harmonic oscillator
We study the eigenvalues E_{n\ell} of the Salpeter Hamiltonian H =
\beta\sqrt(m^2 + p^2) + vr^2, v>0, \beta > 0, in three dimensions. By using
geometrical arguments we show that, for suitable values of P, here provided,
the simple semi-classical formula E = min_{r > 0} {v(P/r)^2 + \beta\sqrt(m^2 +
r^2)} provides both upper and lower energy bounds for all the eigenvalues of
the problem.Comment: 8 pages, 1 figur
Convexity and potential sums for Salpeter-like Hamiltonians
The semirelativistic Hamiltonian H = \beta\sqrt{m^2 + p^2} + V(r), where V(r)
is a central potential in R^3, is concave in p^2 and convex in p. This fact
enables us to obtain complementary energy bounds for the discrete spectrum of
H. By extending the notion of 'kinetic potential' we are able to find general
energy bounds on the ground-state energy E corresponding to potentials with the
form V = sum_{i}a_{i}f^{(i)}(r). In the case of sums of powers and the log
potential, where V(r) = sum_{q\ne 0} a(q) sgn(q)r^q + a(0)ln(r), the bounds can
all be expressed in the semi-classical form E \approx \min_{r}{\beta\sqrt{m^2 +
1/r^2} + sum_{q\ne 0} a(q)sgn(q)(rP(q))^q + a(0)ln(rP(0))}. 'Upper' and 'lower'
P-numbers are provided for q = -1,1,2, and for the log potential q = 0. Some
specific examples are discussed, to show the quality of the bounds.Comment: 21 pages, 4 figure
Coulomb plus power-law potentials in quantum mechanics
We study the discrete spectrum of the Hamiltonian H = -Delta + V(r) for the
Coulomb plus power-law potential V(r)=-1/r+ beta sgn(q)r^q, where beta > 0, q >
-2 and q \ne 0. We show by envelope theory that the discrete eigenvalues
E_{n\ell} of H may be approximated by the semiclassical expression
E_{n\ell}(q) \approx min_{r>0}\{1/r^2-1/(mu r)+ sgn(q) beta(nu r)^q}.
Values of mu and nu are prescribed which yield upper and lower bounds.
Accurate upper bounds are also obtained by use of a trial function of the form,
psi(r)= r^{\ell+1}e^{-(xr)^{q}}. We give detailed results for
V(r) = -1/r + beta r^q, q = 0.5, 1, 2 for n=1, \ell=0,1,2, along with
comparison eigenvalues found by direct numerical methods.Comment: 11 pages, 3 figure
A. Paul Sigurd\u27s Decision
In lieu of an abstract, below is the essay\u27s first paragraph.
Well, son, I don\u27t think anyone really knew how he got it. Some said it was always his and that he was always there. Yet others said that he inherited it from his father. And many be1ieved that it was given to him by an impulsive woman - the Hester Prynne type - who, being in dire straits, had to get rid of it. A few even said that he built it himself when lie was a young man. Me? I never cared how he got it; the fact was that he had it and he was there. But I must confess I always wondered why, I mean with no boats coming into the harbor anymore. And did you know that he used to paint it white every spring? And that he used to put the light on every night? Every night it could be seen from the mainland. Going around and around and around. But why? No boats had come into the harbor for nearly twenty years
Contracting an element from a cocircuit
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Relativistic N-Boson Systems Bound by Oscillator Pair Potentials
We study the lowest energy E of a relativistic system of N identical bosons
bound by harmonic-oscillator pair potentials in three spatial dimensions. In
natural units the system has the semirelativistic ``spinless-Salpeter''
Hamiltonian H = \sum_{i=1}^N \sqrt{m^2 + p_i^2} + \sum_{j>i=1}^N gamma |r_i -
r_j|^2, gamma > 0. We derive the following energy bounds: E(N) = min_{r>0} [N
(m^2 + 2 (N-1) P^2 / (N r^2))^1/2 + N (N-1) gamma r^2 / 2], N \ge 2, where
P=1.376 yields a lower bound and P=3/2 yields an upper bound for all N \ge 2. A
sharper lower bound is given by the function P = P(mu), where mu =
m(N/(gamma(N-1)^2))^(1/3), which makes the formula for E(2) exact: with this
choice of P, the bounds coincide for all N \ge 2 in the Schroedinger limit m
--> infinity.Comment: v2: A scale analysis of P is now included; this leads to revised
energy bounds, which coalesce in the large-m limi
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