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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 $$m$$.

MSC:
 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34B08 Parameter dependent boundary value problems for ordinary differential equations 41A99 Approximations and expansions
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References:
 [1] Bender C.M., Wu, T.T.: Anharmonic oscillator. Phys. Rev. 184, 1231 (1969); Anharmonic oscillator II. Phys. Rev. D7, 1620 (1973) [2] Price P.J. (1954). Proc Phys Soc London 67:383 · Zbl 0055.21202 · doi:10.1088/0370-1298/67/4/410 [3] Turbiner A.V.: ’The problem of spectra in quantum mechanics and the non-linearization procedure: Usp. Fiz. Nauk. 144, 35–78 (1984), Sov. Phys. – Uspekhi 27, 668–694 (1984) (English translation) [4] Caswell W.E. (1979). Accurate energy levels for the anharmonic oscillator and a summable series for the double-well potential in perturbation theory. Ann. Phys. 123:153–184 · doi:10.1016/0003-4916(79)90269-0 [5] Turbiner, A.V.: A new approach to finding levels of energy of bound states in quantum mechanics: convergent perturbation theory. Soviet Phys. – Pisma ZhETF 30, 379–383 (1979), JETP Lett. 30, 352–355 (1979) (English translation) [6] Turbiner, A.V.: Spectral Riemannian surfaces of the Sturm-Liouvile operator and quasi-exactly-solvable problems. Funktsional’nyi Analiz i ego Prilozhenia 22, 92–94 (1988), Soviet Math. – Functional Anal. and its Appl. 22, 163–166 (1988) (English translation); Quantum mechanics: the problems lying between exactly-solvable and non-solvable. Soviet Phys. – ZhETF 94, 33–45 (1988), JETP 67, 230–236 (1988)(English translation); Quasi-exactly-solvable problems and the SL(2,R) group. Comm. Math. Phys. 118, 467–474 (1988)
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