On a differential operator with absent spectrum

TL;DR

This paper investigates a higher-order differential operator with special boundary conditions, providing an example where the spectral problem has no eigenvalues, challenging traditional spectral theory assumptions.

Contribution

It constructs a novel example of a differential operator with degenerate boundary conditions that lacks eigenvalues, expanding understanding of spectral properties.

Findings

Constructed a differential operator with no eigenvalues

Demonstrated the impact of degenerate boundary conditions on spectrum

Challenged existing spectral theory assumptions

Abstract

In this paper, we consider spectral problem for the nth order ordinary differential operator with degenerate boundary conditions. We construct a nontrivial example of boundary value problem which has no eigenvalues.

Equations

$$l(u)=u^{(n)}(x)+\sum _{m=1}^n{p}_m(x)u^{(n-m)}(x)=\lambda u(x)$$

$$B_j(u)=u^{(n-j)}(0)+d(-1{)}^{j+1}{u}^{(n-j)}(1)=0$$

$$l(u)=\lambda u,u_k^{(j)}(1/2)={\delta }_{k,j},$$

$$u_{-}(x)=\sum _{k=0}^{\nu -1}{c}_{2k+1}{u}_{2k+1}(x),u_{+}(x)=\sum _{k=0}^{\nu -2}{c}_{2k}{u}_{2k}(x),$$

$$u_{-}^{(2k)}(1/2)=0,u_{+}^{(2k+1)}(1/2)=0,k=0,\dots ,\nu -1.$$

$$u_{-}^{(n-j)}(0)+(-1{)}^{j+1}{u}_{-}^{(n-j)}(1)=0,u_{+}^{(n-j)}(0)+(-1{)}^j{u}_{+}^{(n-j)}(1)=0$$

$$u_{-}^{(n-j)}(0)+u_{+}^{(n-j)}(0)+(-1{)}^{j+1}d(u_{-}^{(n-j)}(1)+u_{+}^{(n-j)}(1))=0$$

$$u_{-}^{(n-j)}(0)(1-d)+u_{+}^{(n-j)}(0)(1+d)=0$$

$$(1+d)\sum _{k=0}^{\nu -2}{c}_{2k}{u}_{2k}^{(n-j)}(0)+(1-d)\sum _{k=0}^{\nu -1}{c}_{2k+1}{u}_{2k+1}^{(n-j)}(0)=0$$

$$\sum _{i=0}^{n-1}{c}_i(1+d(-1{)}^i)u_i^{(n-j)}(0)=0$$

$$\Delta =(1-d^2{)}^{\nu }\det ||u_i^{(n-j)}(0)||.$$

Full text

Consider the boundary value problem for the $n$th order ordinary differential equation

[TABLE]

with boundary conditions

[TABLE]

$(j=1,\ldots n)$, where $n=2\nu$, coefficient $d$ is an arbitrary complex number, and the complex-valued coefficients $p_{m}(x)$ are functions in the class $L_{1}(0,1)$. Suppose that $p_{m}(x)=(-1)^{m}p_{m}(1-x)$ almost everywhere on the segment $[0,1]$, $m=1,\ldots,n$. We will study the spectrum of problem (1), (2).

Let a function $u_{k}(x)$ be the solution of the Cauchy problem

[TABLE]

where $k=0,\ldots,n-1$, $j=0,\ldots,n-1$. Denote

[TABLE]

where $c_{i}$ are arbitrary constants $(i=0,\ldots,n-1)$. Then

[TABLE]

Obviously, that the functions $w_{-}(x)=-u_{-}(1-x)$ and $w_{+}(x)=u_{+}(1-x)$ are the solutions of equation (1) and satisfy the same initial conditions at the point $1/2$ just as the functions $u_{-}(x)$ and $u_{+}(x)$, correspondingly. This, together with the uniqueness of the solution of Cauchy problem (3) implies, that $u_{-}(x)=-u_{-}(1-x),\quad u_{+}(x)=u_{+}(1-x)$ for $0\leq x\leq 1$. It follows from this that

[TABLE]

$(j=1,\ldots,n)$. It follows from (4) that for any complex number $\lambda$ the function $u_{-}(x)$ is a solution of problem (1), (2) if $d=1$, and for any complex number $\lambda$ the function $u_{+}(x)$ is a solution of problem (1), (2) if $d=-1$. Thus, we establish that if $d=\pm 1$ the spectrum of problem (1), (2) fills all complex plane. If $p_{m}(x)\in C^{n-m}(0,1)$, $m=1,\ldots,n$, this assertion was proved in [1].

Assume, for a number $\lambda$ a function $\tilde{u}(x)$ is a solution of problem (1), (2) if $d\neq\pm 1$. Then $\tilde{u}(x)=u_{+}(x)+u_{-}(x)$. We see that

[TABLE]

$(j=1,\ldots,n)$. It follows from (4), (5) that

[TABLE]

$(j=1,\ldots,n)$. From this and the definition of the functions $u_{+}(x)$ and $u_{-}(x)$, we have

[TABLE]

$(j=1,\ldots,n)$, hence, the constants $c_{i}$ $(i=0,\ldots,n-1)$ satisfy the system of linear equations

[TABLE]

$(j=1,\ldots,n)$. The determinant of linear system (6) is

[TABLE]

Since the last determinant is the Wronskian of the fundamental system of the solutions of equation (1), it is nonzero. Therefore, system (6) has only trivial solution, i.e. the function $\tilde{u}(x)\equiv 0$. Hence, if $d\neq\pm 1$ problem (1), (2) has no eigenvalues.

For the first time problem (1), (2) for $n=2$ was investigated in [2].

References

1. V.A. Sadovnichy, B.E. Kanguzhin. On a connection between the spectrum of a differential operator with symmetric coefficients and boundary conditions. Dokl. Akad. Nauk SSSR. 1982. V. 267. No. 2. P. 310-313.

2. M.H. Stone. Irregular differential systems of order two and the related expansion problems. Trans. Amer. Math. Soc. 1927. V. 29. No. 1. P. 23-53.

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