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Anharmonic oscillator and double-well potential: Approximating eigenfunctions. (English) Zbl 1092.34049
The author notes that in the last 50 years the one-dimensional anharmonic oscillator (meaning \(m^2\geq 0)\), the corresponding Hamiltonian \[ {\mathbf H}=-d^2/dx^2+m^2x^2+gx^4\tag{1} \] and the Schrödinger equation \[ {\mathbf H}\Psi = E(m^2,g)\psi\tag{2} \] have been intensively studied and resulted in nontrivial applications to quantum mechanics, and to quantum field theory. Here, the author develops a uniform approximation of the ground state eigenfunction of (1), which remains valid for different values of the coupling constant \(g\) and the parameter \(m\).

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B08 Parameter dependent boundary value problems for ordinary differential equations
41A99 Approximations and expansions
Full Text: DOI arXiv
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