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This may give you more information than you need, but this perturbed harmonic oscillator can be solved in terms of parabolic cylinder functions. The eigenvalues are given by Equation 3.19 of The energy level structure of a variety of one-dimensional confining potentials and the effects of a local singular perturbation. The lowest eigenvalue $\epsilon_0$ is the smallest $\epsilon$ that solves

$$\alpha D_{\sigma-1/2}(\alpha)=2D_{\sigma+1/2}(\alpha),\;\;\sigma=\epsilon+\alpha^2/4.$$

For small $\alpha$ perturbation theory gives a linear growth in $\alpha$ of $\epsilon_0$. For large $\alpha$ the quadratic part of the potential can be neglected and $\epsilon_0$ grows more slowly as $\alpha^{2/3}$. Figure 8 in the cited paper gives a plot.

This may give you more information than you need, but this perturbed harmonic oscillator can be solved in terms of parabolic cylinder functions. The eigenvalues are given by Equation 3.19 of The energy level structure of a variety of one-dimensional confining potentials and the effects of a local singular perturbation. The lowest eigenvalue $\lambda_1$ is the smallest $\lambda$ that solves

$$\epsilon D_{\sigma-1/2}(\epsilon)=2D_{\sigma+1/2}(\epsilon),\;\;\sigma=\lambda+\epsilon^2/4.$$

For small $\epsilon$ perturbation theory gives a linear growth in $\epsilon$ of $\lambda_1$. For large $\epsilon$ the quadratic part of the potential can be neglected and $\lambda_1$ grows more slowly as $\epsilon^{2/3}$. Figure 8 in the cited paper gives a plot.

This may give you more information than you need, but this perturbed harmonic oscillator can be solved in terms of parabolic cylinder functions. The eigenvalues are given by Equation 3.19 of The energy level structure of a variety of one-dimensional confining potentials and the effects of a local singular perturbation. The lowest eigenvalue is the smallest $\epsilon$ that solves

$$\alpha D_{\sigma-1/2}(\alpha)=2D_{\sigma+1/2}(\alpha),\;\;\sigma=\epsilon+\alpha^2/4.$$

This may give you more information than you need, but this perturbed harmonic oscillator can be solved in terms of parabolic cylinder functions. The eigenvalues are given by Equation 3.19 of The energy level structure of a variety of one-dimensional confining potentials and the effects of a local singular perturbation. The lowest eigenvalue $\epsilon_0$ is the smallest $\epsilon$ that solves

$$\alpha D_{\sigma-1/2}(\alpha)=2D_{\sigma+1/2}(\alpha),\;\;\sigma=\epsilon+\alpha^2/4.$$

For small $\alpha$ perturbation theory gives a linear growth in $\alpha$ of $\epsilon_0$. For large $\alpha$ the quadratic part of the potential can be neglected and $\epsilon_0$ grows more slowly as $\alpha^{2/3}$. Figure 8 in the cited paper gives a plot.

This may give you more information than you need, but this perturbed harmonic oscillator can be solved in terms of parabolic cylinder functions. The eigenvalues are given by Equation 3.19 of The energy level structure of a variety of one-dimensional confining potentials and the effects of a local singular perturbation.

This may give you more information than you need, but this perturbed harmonic oscillator can be solved in terms of parabolic cylinder functions. The eigenvalues are given by Equation 3.19 of The energy level structure of a variety of one-dimensional confining potentials and the effects of a local singular perturbation. The lowest eigenvalue is the smallest $\epsilon$ that solves

$$\alpha D_{\sigma-1/2}(\alpha)=2D_{\sigma+1/2}(\alpha),\;\;\sigma=\epsilon+\alpha^2/4.$$