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Analytic approximants for the eigenvalues of the one-dimensionalSchr\"odinger equation with potentials of the form $V(x)= Ax^a + Bx^b$ arefound using a multi-point quasi-rational approximation technique. Thistechnique is based on the use of the power series and asymptotic expansion ofthe eigenvalues in $\lambda=A^{-\frac{b+2}{a+2}}B$, as well as the expansion atintermediate points. These expansions are found through a system ofdifferential equations. The approximants found are valid and accurate for anyvalue of $\lambda$. As examples, the technique is applied to the quartic andsextic anharmonic oscillators.

Multi-point quasi-rational approximants for the energy eigenvalues of potentials of the form V(x)= Ax^a + Bx^b