Zeros of eigenfunctions of some anharmonic oscillators

where P is a real even polynomial with positive leading coefficient, which is called a potential. The boundary condition is equivalent to y ∈ L(R) in this case. It is well-known that the spectrum is discrete, and all eigenvalues λ are real and simple, see, for example [3, 14]. The spectrum can be arranged in an increasing sequence λ0 < λ1 < . . .. Eigenfunctions y are real entire functions of order (deg P + 2)/2 and each of them has finitely many real zeros. The number of real zeros of an eigenfunction is equal to the subscript of the corresponding eigenvalue λk. Asymptotic behavior of complex zeros of eigenfunctions is well-known, in particular, their arguments accumulate to finitely many directions, the socalled Stokes’ directions [1, 2]. Using this one can show that for a real even potential P of degree 4 with positive leading coefficient, all but finitely many zeros of each eigenfunction lie on the imaginary axis. See also [17] where a similar result was obtained for some cubic potentials.