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The generalized d’Alembert operator on compactified pseudo-Euclidean space. (English. Russian original) Zbl 1196.47035

Math. Notes 85, No. 5, 630-637 (2009); translation from Mat. Zametki 85, No. 5, 652-660 (2009).
The author considers the operator \[ D = R(x) \left(\sum^n_{j=1} \sigma_i{\partial^2\over\partial x^2_i}\right) R(x)\quad\text{on }\mathbb{R}^n, \] where \(\sigma+i= 1\) for \(i= 1,2,\dots,p\), and \(\sigma_i= -1\) for \(i= p+1,\dots, n\), with \(0< p< n\), \(q= n-p\), and \(n\) being assumed even, while \(R\) is a certain rational function defined in the paper. It is shown that under some assumptions on the behaviour of \(R\) at \(\infty\), the operator \(D\) is selfadjoint on \(L_2(\mathbb{R}^n)\), while its closure \(\overline D\) has the eigenvalues \[ \lambda_{k,l}= \Biggl(l+{q-1\over 2}\Biggr)^2-\Biggl(k+{p-1\over 2}\Biggr)^2. \] The eigenfunctions are also explicitly determined in terms of spherical functions.

MSC:

47F05 General theory of partial differential operators
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35L10 Second-order hyperbolic equations
58J45 Hyperbolic equations on manifolds
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