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On quantum fields satisfying a given wave equation. (English) Zbl 0777.35064
The authors treat here the quantum fields on $$\mathbb{R}\times S^ 1$$ satisfying the nonlinear Klein-Gordon equation $(\square+m^ 2)\varphi+\lambda:P'(\varphi):_ v=0. \tag{1}$ Here $$P$$ is a given real polynomial, bounded below, and $$:P'(\varphi):_ v$$ means the one by the renormalized powers developed by I. Segal [J. Funct. Anal. 4, 404- 456 (1969; Zbl 0187.392); 6, 29-75 (1970; Zbl 0202.422)]. In order to solve the Eq. (1), the study of the ground state, or vacuum of the self- adjoint operator $H(\lambda P,v)=H_ 0+\int_{t=0}:\lambda P(\varphi):_ vdx$ is required. If the vacuum is equal to v itself, up to a scalar multiple, $\varphi_{int}(t,x)=e^{itH(\lambda P,v)}\varphi(0,x)e^{-itH(\lambda P,v)}$ is a solution of the Eq. (1). The state $$u(\lambda)$$, equal to the vacuum of $$H(\lambda P,u(\lambda))$$ for $$\lambda\in[0,\lambda_ 0]$$, is uniquely defined under the two conditions “norm-continuous” and “boundedness”. When $$P(x)=x^ 4$$, the authors give two different states $$u_ i(\lambda)$$, $$i=1,2$$, equal to the vacua of $$H(\lambda P,u_ i(\lambda))$$ for $$\lambda\in(0,\lambda_ 0]$$, respectively. Here $$H(\lambda P,u_ i(\lambda))$$, $$i=1,2$$, are not unitary equivalent. Finally they show the similar known results for the one dimensional anharmonic oscillator.
Reviewer: H.Yamagata (Osaka)

##### MSC:
 35Q40 PDEs in connection with quantum mechanics 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35L75 Higher-order nonlinear hyperbolic equations 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
##### Keywords:
existence; non-uniqueness; nonlinear Klein-Gordon equation
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