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Analytic continuation of eigenvalues of a quartic oscillator. (English) Zbl 1184.34083
For the boundary value problem on the real line
\[ -y''(x)+(x^4+\alpha x^2)y(x)=\lambda y(x),\quad y(-\infty)=y(\infty)=0 \]
the authors study the analytic continuation of its eigenvalues depending on the complex parameter \(\alpha\). Denoting by \(\lambda_n\) the eigenvalues which are real analytic functions of \(\alpha>0\), the authors prove their main result: a) all \(\lambda_n\) are branches of two multi-valued analytic functions \(\Lambda^i\), \(i=0,1\), of \(\alpha\), one for even \(n\), another for odd \(n\); b) the only singularities of \(\Lambda^i\) over the \(\alpha\)-plane are algebraic ramification points; c) for every bounded set \(X\) in the \(\alpha\)-plane, there are only finitely many ramification points of \(\Lambda^i\) over \(X\). At the end of the paper, the authors briefly mention some other one-parametric families of linear differential operators with polynomial potentials which can be treated by the same method.

MSC:
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34B09 Boundary eigenvalue problems for ordinary differential equations
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