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Spectrum of a differential operator with an infinite number of variables. (English. Russian original) Zbl 0545.47030

Sib. Math. J. 23, 880-890 (1983); translation from Sib. Mat. Zh. 23, No. 6, 147-159 (1982).
Let \(\gamma_ k(x)=\sqrt{\frac{\epsilon_ k}{\pi}e^{-\epsilon_ kx^ 2_ k}}, \epsilon_ k>0, \sum^{\infty}_{1}\epsilon_ k<\infty, k=1,2,..\). and \(dg_{\epsilon}(x)=d\gamma_ 1(x_ 1)\otimes d\gamma_ 2(x_ 2)\otimes...\) be the measure on \({\mathbb{R}}^{\infty}=\{x=(x_ 1x_ 2,...),x_ k\in {\mathbb{R}}\}; L_ 2({\mathbb{R}}^{\infty},dg_{\epsilon}(x))\) denotes the space of square- integrable functions on \({\mathbb{R}}^{\infty}\) with measure \(dg_{\epsilon}(x)\). The authors investigate the spectra of the closure in \(L_ 2({\mathbb{R}}^{\infty},dg_{\epsilon}(x))\) of the following differential operator \[ (*)\quad Au(x)=-\sum^{\infty}_{1}D^ 2_ ku(x)+q(x)u(x)\quad D_ ku(x)=\frac{1}{\sqrt{\gamma_ k(x_ k)}}\frac{\partial}{\partial x_ k}\sqrt{\gamma_ k(x_ k)}u(x) \] similar to the Schrödinger operator; here u(x) belongs to the linear span of cylindrical functions \(u(x)=u(x_ 1,x_ 2,..,x_ p)\in C_ 0^{\infty}({\mathbb{R}}^ p), p=p(u)\), and \(q(x)\in L_ 2({\mathbb{R}}^{\infty},dg_{\epsilon}(x))\) is real-valued semi-bounded from below (this provides the self-adjointness of A; cf. Ju. Berezanskij, V. Samojlenko, Dopovidi Akad. Nauk Ukr. RSR, Ser. A 1979, 691-695 (1979; Zbl 0415.47022).
The investigation of the spectra of (*) is tightly connected with Sobolev’s imbedding theorems; but they are different in the case of \({\mathbb{R}}^{\infty}\) (than in \({\mathbb{R}}^ N)\) and this provides a quite different picture of the spectra of A. Operators (*) arise from the quantum field theory and were investigated earlier in the case of special potentials only [cf. J. Glimm, A. Jaffe, Ann. Math. II. Ser. 91, 362-401 (1970; Zbl 0191.270) and M. Reed, Commun. Math. Phys. 11, 346-357 (1969)].
Reviewer: R.Dudučava

MSC:

47F05 General theory of partial differential operators
47A10 Spectrum, resolvent
81T08 Constructive quantum field theory
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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[1] Yu. M. Berezanskii and V. G. Samoilenko, ?On self-adjoint differential operators with a finite and infinite numbers of variables,? Dokl. Akad. Nauk Ukr. SSR, No. 9, 691-696 (1979).
[2] J. Glimm and A. Jaffe, ?The ?(?4)2 quantum field theory without cutoffs. II. The field operators and the approximate vacuum,? Ann. Math.,91, 362-401 (1970). · Zbl 0191.27005 · doi:10.2307/1970582
[3] R. Höegh-Krohn, ?On the spectrum of the space cutoff:P(?): Hamiltonian in two space-time dimensions,? Comm. Math. Phys.,21, 256-260 (1971). · doi:10.1007/BF01647123
[4] M. Reed, ?The damped self-interaction,? Comm. Math. Phys.,11, No. 4, 346-357 (1969). · doi:10.1007/BF01645855
[5] S. L. Sobolev, Introduction into the Theory of Cubature Formulas [in Russian], Nauka, Moscow (1974).
[6] M. Sh. Birman, ?On the spectrum of singular boundary value problems,? Mat. Sb.,95, No. 1, 108-129 (1974).
[7] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I. Functional Analasis, Academic Press, New York, London (1972). · Zbl 0242.46001
[8] I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators [in Russian], Fizmatgiz, Moscow (1963). · Zbl 0143.36505
[9] G. E. Shilov and Fan Dik Tin, Integral, Measure, and Derivative on Linear Spaces [in Russian], Nauka, Moscow (1967).
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