Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 327 / 2 / b

Solution

For each one-dimensional factor

P_{j} (D_{j})

E_{j} = H (x_{j}) g_{j} (x_{j})

g_{j} (x_{j}) = \sum_{r = 1}^{N_{j}} α_{r j} (x_{j}) e^{β_{r j} x_{j}},

where each

α_{r j}

is a polynomial whose degree is one less than the multiplicity of the associated root. Constants, including powers of

i

from

D = - i \partial

, can be absorbed into the polynomials and exponents.

E (x_{1}, \dots, x_{n}) = \prod_{j = 1}^{n} E_{j} (x_{j}) .

It vanishes unless every

x_{j} > 0

, has the required polynomial-exponential form there, and satisfies

P (D) E = \prod_{j = 1}^{n} P_{j} (D_{j}) E_{j} = δ_{0} (x_{1}) \otimes \dots \otimes δ_{0} (x_{n}) = δ_{0} .

Solved by gpt-5.6-sol high.

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