Hermitizable, isospectral complex second-order differential operators
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Hermitizable, isospectral complex second-order differential operators Mu-Fa CHEN1,2,3 , Jin-Yu LI2 1 Research Institute of Mathematical Science, Jiangsu Normal University, Xuzhou 221116, China 2 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China 3 Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of Education, Beijing 100875, China
c Higher Education Press 2020
Abstract The first aim of the paper is to study the Hermitizability of secondorder differential operators, and then the corresponding isospectral operators. The explicit criteria for the Hermitizable or isospectral properties are presented. The second aim of the paper is to study a non-Hermitian model, which is now well known. In a regular sense, the model does not belong to the class of Hermitizable operators studied in this paper, but we will use the theory developed in the past years, to present an alternative and illustrated proof of the discreteness of its spectrum. The harmonic function plays a critical role in the study of spectrum. Two constructions of the function are presented. The required conclusion for the discrete spectrum is proved by some comparison technique. Keywords Hermitizable, isospectral, differential operators, non-Hermitian model, discrete spectrum MSC2020 47B15, 47B91, 47B95 1
Introduction
Denote by C m (Rd ) the set of functions on Rd with continuous derivatives up to order m. Let a = (aij (x))di,j=1 and b = (bi (x))di=1 be given complex matrix and vector on Rd , respectively. Assume aij ∈ C 1 (Rd , C) for each i, j. Next, let V ∈ C 1 (Rd , R) and dµ = eV dx. Thus, the first part of the paper is an extension of [5; §5] replacing the Lebesgue measure by µ. Consider the following complex
Received June 1, 2020; accepted September 3, 2020 Corresponding author: Mu-Fa CHEN, E-mail: [email protected]
2
Mu-Fa CHEN, Jin-Yu LI
second-order differential operator X X L= ∂i (aij ∂j ) + bi ∂i − c, i,j
(1)
i
where ∂i = d/dxi and c ∈ L2 (µ). We say that L with domain D(L) is Hermitizable with respect to the measure µ if L is a Rself-adjoint operator on the complex space L2 (µ) with inner product (f, g)µ = f gdµ : f, g ∈ D(L) ⊂ L2 (µ).
(Lf, g)µ = (f, Lg)µ ,
m For vectors F = {fk }m k=1 and G = {gk }k=1 , set
hF, Giµ =
m X
(fk , gk )µ .
k=1
We have the following result for the Hermitizability. The main part of the result is given in [9]. An alternative proof of the result is delayed to Section 4 of the paper. Theorem 1 The operator L is Hermitizable with respect to the measure µ if and only if a is Hermitian (i.e., aH := a∗ = a) and Re b = (Re a)(∂V ),
(2)
2 Im c = −((∂V )∗ + ∂ ∗ )((Im a)(∂V ) + Im b),
(3)
where x∗ denote the transpose of x. If so, its quadratic form is as follows: √ (−Lf, g)µ = ha∂f, ∂giµ + −1 h(Im b + (Im a)(∂V ))f, ∂giµ + (cf, g)µ , f, g ∈ D(L) ⊂ L2 (µ), (4) where Im c satisfies (3). We now make a remark on the ordinary form of the second-order differential operator. Noting that 2 ∂i (aij ∂j f ) = aij ∂ij f + (∂i aij )∂j f,
we have X
∂i (ai
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