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.


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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