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An explicit asymptotic formula is obtained for the norms of the spectral projections of the non-self-adjoint harmonic oscillator $H$. It is deduced that the spectral expansion of $\rme^{-Ht}$ is norm convergent if and only if $t$ is greater than a certain explicit positive constant.

Necessary and sufficient conditions are found for the existence of two positive solutions of the semilinear elliptic equation Δu + q(|x|)u = f(x, u) in an exterior domain Ω⊂ℝn, n ≥ 1, where q, f are real-valued and locally Hölder continuous, and f(x, u) is nonincreasing in u for each fixed x∈Ω. An example is the singular stationary Klein-Gordon equation Δu — k 2 u = p(x)u -λ where k and λ are positive constants. In this case NASC are given for the existence of two positive solutions u i (x) in some exterior subdomain of Ω such that both |x|m exp[(-l)i-1 k|x|]u i (x) are bounded and bounded away from zero in this subdomain, m = (n —1)/2, i = 1, 2.

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SPECTRAL ASYMPTOTICS OF THE NON-SELF-ADJOINT HARMONIC OSCILLATOR

Asymptotic Theory of Singular Semilinear Elliptic Equations

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2 results

SPECTRAL ASYMPTOTICS OF THE NON-SELF-ADJOINT HARMONIC OSCILLATOR

Asymptotic Theory of Singular Semilinear Elliptic Equations