This paper reports a high-resolution experimental study and a numerical analysis of the Rb2 6 1 Σ + g ion-pair state. Large number of ro-vibrational term values spanning a wide range of the rotational and vibrational quantum numbers were measured using the optical-optical double resonance technique. The set of term values was simulated with a model of a piece-wise multi-parameter potential energy function based on the generalized splines. This function reproduces the experimental data with reasonable accuracy and in addition allows us to incorporate in the potential function the non-trivial features at longer internuclear range, such as multiple wells, predicted by the ab initio calculations.
This paper reports observations and analysis of the Rb2 31Πg state. A total of 323 rovibrational term values spanning the range of the rotational quantum number J = 7 through 77 and the vibrational quantum number v = 2 through 23 (about 1/3 of the potential well depth) were measured using the optical–optical double resonance technique. The term values are simulated within a model of a piece-wise multi-parameter potential energy function based on the generalized splines. This function not only enables a reproduction of the experimental data with a reasonable quality but also approximates the available ab initio function in its whole range with a uniform accuracy.
In analyzing finite-state Markov chains knowing the exact eigenvalues of the transition probability matrix
P
P
is important information for predicting the explicit transient behavior of the system. Once the eigenvalues of
P
P
are known, linear algebra and duality theory are used to find
P
k
P^{k}
where
k
=
2
,
3
,
4
,
…
k= 2,3,4,\ldots
. This article is about finding explicit eigenvalue formulas, that scale up with the dimension of
P
P
for various Markov chains. Eigenvalue formulas and expressions of
P
k
P^{k}
are first presented when
P
P
is tridiagonal and Toeplitz. These results are generalized to tridiagonal matrices with alternating birth-death probabilities. More general eigenvalue formulas and expression of
P
k
P^{k}
are obtained for non-tridiagonal transition matrices
P
P
that have both catastrophe-like and birth-death transitions. Similar results for circulant matrices are also explored. Applications include finding probabilities of sample paths restricted to a strip and generalized ballot box problems. These results generalize to Markov processes with
P
k
P^{k}
being replaced by
e
Q
t
e^{Qt}
where
Q
Q
is a transition rate matrix.
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