Near-Dirichlet quantum dynamics for a $p^3$-corrected particle on an interval

Abstract

We study a nonrelativistic quantum mechanical particle on an interval of finite length with a Hamiltonian that has a $p^3$ correction term, modelling potential low energy quantum gravity effects. We describe explicitly the $U(3)$ family of the self-adjoint extensions of the Hamiltonian and discuss several subfamilies of interest. As the main result, we find a family of self-adjoint Hamiltonians, indexed by four continuous parameters and one binary parameter, whose spectrum and eigenfunctions are perturbatively close to those of the uncorrected particle with Dirichlet boundary conditions, even though the Dirichlet condition as such is not in the $U(3)$ family. Our boundary conditions do not single out distinguished discrete values for the length of the interval in terms of the underlying quantum gravity scale.

Appendices

Appendix 1: Subspaces of self-adjointness

In this appendix we perform the maximal linear subspace analysis that leads to the self-adjointness boundary conditions (2.8) in the main text.

1.1 Preliminaries

Let $n$ be a positive integer and $\mathcal {H} = \mathbb {C}^{2n}$. Define on $\mathcal {H}$ the Hermitian form

$$\begin{aligned} B(u,v) = u^\dagger \begin{pmatrix} I &{}\quad 0 \\ 0 &{}\quad -I \end{pmatrix} v, \end{aligned}$$

(6.1)

where $I$ is the $n\times n$ identity matrix.

Lemma

The maximal linear subspaces $V \subset \mathcal {H}$ on which $B(u,v)=0$ for all $u,v \in V$ are

$$\begin{aligned} V_U = \bigl \{ v \in \mathcal {H} \mid \bigl ( {\begin{matrix} U &{}\quad -I \\ 0 &{}\quad 0 \end{matrix}} \bigr ) v =0 \bigr \} , \end{aligned}$$

(6.2)

where $U \in U(n)$.

Proof

Let $V \subset \mathcal {H}$ be a linear subspace on which $B(u,v)=0$ for all $u,v \in V$. Suppose $w = \bigl ( {\begin{matrix} w_1 \\ w_2 \end{matrix}} \bigr ) \in V$ where $w_1, w_2 \in \mathbb {C}^n$. Then $B(w,w)=0$ implies $\Vert w_1\Vert = \Vert w_2\Vert$. As $V$ is a linear subspace, each $v \in V$ must hence have the form $\bigl ( {\begin{matrix} v_1 \\ U v_1 \end{matrix}} \bigr )$, where $U$ is a constant $n\times n$ matrix, such that if $V_1 \subset \mathbb {C}^n$ denotes the projection of $V$ to its first $n$ components, $U$ maps $V_1$ isometrically to $\mathbb {C}^n$. For $u = \bigl ( {\begin{matrix} u_1 \\ U u_1 \end{matrix}} \bigr )$ and $v = \bigl ( {\begin{matrix} v_1 \\ U v_1 \end{matrix}} \bigr )$ in $V$, $B(u,v)=0$ is equivalent to $u_1^\dagger \bigl (U^\dagger U - I \bigr ) v_1 = 0$. This holds for all $u_1 , v_1 \in \mathbb {C}^n$ iff $U^\dagger U = I$. $\square$

Remark

The maximal linear subspaces on which $B(v,v)=0$ coincide with (6.2). The proof is as above but setting at every step $u=v$.

For generalisations, see [17, 18].

1.2 Main proposition

Let $n$ be a positive integer and $\mathcal {H} = \mathbb {C}^{2n}$. Define on $\mathcal {H}$ the Hermitian form

$$\begin{aligned} C(u,v) = u^\dagger A v, \end{aligned}$$

(6.3)

where $A$ is a Hermitian $2n\times 2n$ matrix with $n$ strictly positive eigenvalues and $n$ strictly negative eigenvalues (each eigenvalue counted with its multiplicity). By matrix diagonalisation, there exists a unitary $2n\times 2n$ matrix $P$ and a real diagonal positive definite $2n\times 2n$ matrix $D$ such that

$$\begin{aligned} A = {(D P)}^\dagger \begin{pmatrix} I&{}\quad 0 \\ 0 &{}\quad -I \end{pmatrix} (D P) . \end{aligned}$$

(6.4)

Proposition

The maximal linear subspaces $V \subset \mathcal {H}$ on which $C(u,v)=0$ for all $u,v \in V$ are

$$\begin{aligned} V_U = \bigl \{ v \in \mathcal {H} \mid \bigl ( {\begin{matrix} U &{}\quad -I \\ 0 &{}\quad 0 \end{matrix}} \bigr ) (D P) v =0 \bigr \} , \end{aligned}$$

(6.5)

where $U \in U(n)$.

Proof

Follows from the Lemma by observing that $C(u,v) = B( DP u, D P v)$. $\square$

1.3 Application

We specialise (6.3) to

$$\begin{aligned} A = \begin{pmatrix} G&{}\quad 0 \\ 0 &{}\quad -G \end{pmatrix} \end{aligned}$$

(6.6)

where $G$ is a Hermitian $3\times 3$ matrix with the eigenvalues $\lambda _-<0$, $\lambda _+>0$ and $\lambda _0>0$ and the corresponding orthogonal normalised eigen-covectors

$$\begin{aligned} \begin{pmatrix} a_1&a_2&a_3 \end{pmatrix}, \quad \begin{pmatrix} b_1&b_2&b_3 \end{pmatrix}, \quad \begin{pmatrix} c_1&c_2&c_3 \end{pmatrix}. \end{aligned}$$

(6.7)

The matrix

$$\begin{aligned} \tilde{P} = \begin{pmatrix} a_1&{}\quad a_2 &{}\quad a_3 \\ b_1&{}\quad b_2 &{}\quad b_3 \\ c_1&{}\quad c_2 &{}\quad c_3 \\ \end{pmatrix} \end{aligned}$$

(6.8)

is then unitary, and

$$\begin{aligned} \begin{pmatrix} \tilde{P}&{} 0 \\ 0 &{} \tilde{P} \end{pmatrix} A \begin{pmatrix} \tilde{P}^\dagger &{} 0 \\ 0 &{} \tilde{P}^\dagger \end{pmatrix} = {{\mathrm{diag}}}(\lambda _-, \lambda _+, \lambda _0,- \lambda _-, - \lambda _+, - \lambda _0). \end{aligned}$$

(6.9)

Let

$$\begin{aligned} Q= & {} \begin{pmatrix} 0&{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0&{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0&{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 1&{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0&{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0&{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ \end{pmatrix}, \end{aligned}$$

(6.10)

$$\begin{aligned} \tilde{D}= & {} \begin{pmatrix} \sqrt{-\lambda _-}&{}\quad 0 &{}\quad 0 \\ 0&{}\quad \sqrt{\lambda _+} &{}\quad 0 \\ 0&{}\quad 0 &{}\quad \sqrt{\lambda _0} \\ \end{pmatrix}. \end{aligned}$$

(6.11)

Then (6.4) holds with

$$\begin{aligned} DP = \begin{pmatrix} \tilde{D}&{}\quad 0 \\ 0 &{}\quad \tilde{D} \end{pmatrix} Q \begin{pmatrix} \tilde{P}&{}\quad 0 \\ 0 &{}\quad \tilde{P} \end{pmatrix}. \end{aligned}$$

(6.12)

Writing in (6.5)

$$\begin{aligned} v = \begin{pmatrix} \rho _1 \\ \rho _2 \\ \rho _3 \\ \sigma _1 \\ \sigma _2 \\ \sigma _3 \\ \end{pmatrix}, \end{aligned}$$

(6.13)

we have

$$\begin{aligned} DP \begin{pmatrix} \rho _1 \\ \rho _2 \\ \rho _3 \\ \sigma _1 \\ \sigma _2 \\ \sigma _3 \\ \end{pmatrix}= \begin{pmatrix} \sqrt{-\lambda _-} \left( a_1 \sigma _1 + a_2 \sigma _2 + a_3 \sigma _3\right) \\ \sqrt{\lambda _+} \left( b_1 \rho _1 + b_2 \rho _2 + b_3 \rho _3\right) \\ \sqrt{\lambda _0} \left( c_1 \rho _1 + c_2 \rho _2 + c_3 \rho _3\right) \\ \sqrt{-\lambda _-} \left( a_1 \rho _1 + a_2 \rho _2 + a_3 \rho _3\right) \\ \sqrt{\lambda _+} \left( b_1 \sigma _1 + b_2 \sigma _2 + b_3 \sigma _3\right) \\ \sqrt{\lambda _0} \left( c_1 \sigma _1 + c_2 \sigma _2 + c_3 \sigma _3\right) \\ \end{pmatrix}, \end{aligned}$$

(6.14)

and the subspace condition (6.5) reads

$$\begin{aligned} U \begin{pmatrix} \sqrt{-\lambda _-} \left( a_1 \sigma _1 + a_2 \sigma _2 + a_3 \sigma _3\right) \\ \sqrt{\lambda _+} \left( b_1 \rho _1 + b_2 \rho _2 + b_3 \rho _3\right) \\ \sqrt{\lambda _0} \left( c_1 \rho _1 + c_2 \rho _2 + c_3 \rho _3\right) \\ \end{pmatrix} = \begin{pmatrix} \sqrt{-\lambda _-} \left( a_1 \rho _1 + a_2 \rho _2 + a_3 \rho _3\right) \\ \sqrt{\lambda _+} \left( b_1 \sigma _1 + b_2 \sigma _2 + b_3 \sigma _3\right) \\ \sqrt{\lambda _0} \left( c_1 \sigma _1 + c_2 \sigma _2 + c_3 \sigma _3\right) \\ \end{pmatrix}. \end{aligned}$$

(6.15)

This is the condition (2.8) in the main text.

Appendix 2: Small $q$ expansions of the eigenvalues and eigen-covectors

In this appendix we give the small $q$ expansions of the eigenvalues and $\sqrt{|\lambda |}$ times the normalised eigen-covectors (2.6) of the matrix (2.3). The phases of the eigen-covectors are chosen so that $a_1>0$, $b_1>0$ and $c_3>0$.

$$\begin{aligned} \lambda _-= & {} -1+\frac{1}{2}\,q-\frac{5}{8}\,{q}^{2}-\frac{1}{2}\,{q}^{3}-{\frac{7}{128}}\,{q}^{4}+\frac{1}{2}\,{q} ^{5}+{\frac{675}{1024}}\,{q}^{6} + O({q}^8) \end{aligned}$$

(6.16a)

$$\begin{aligned} \lambda _+= & {} 1+\frac{1}{2}\,q+\frac{5}{8}\,{q}^{2}-\frac{1}{2}\,{q}^{3}+{\frac{7}{128}}\,{q}^{4}+\frac{1}{2}\,{q}^ {5}-{\frac{675}{1024}}\,{q}^{6} + O({q}^8) \end{aligned}$$

(6.16b)

$$\begin{aligned} \lambda _0= & {} {q}^{3} \left( 1-{q}^{2}+3\,{q}^{6} + O({q}^8) \right) \end{aligned}$$

(6.16c)

$$\begin{aligned} \sqrt{-\lambda _-} \,a_1= & {} \frac{1}{\sqrt{2}} \Bigl ( 1+\frac{3}{16}\,{q}^{2}-{\frac{83}{512}}\,{q}^{4}+{ \frac{3605}{8192}}\,{q}^{6}+\frac{1}{2}\,{q}^{7} + O({q}^8) \Bigr ) \end{aligned}$$

(6.17a)

$$\begin{aligned} \sqrt{-\lambda _-} \,a_2= & {} -\frac{i}{\sqrt{2}} \Bigl ( 1 -\frac{1}{2}\,q -\frac{3}{16}\,{q}^{2} -{\frac{3}{32}}\,{q}^{3} +{\frac{101}{512}}\,{q}^{4} +{\frac{595}{1024}}\,{q}^{5} +{\frac{4035}{8192}}\,{q}^{6} \nonumber \\&-{\frac{10261}{16384}}\,{q}^{7} + O({q}^8) \Bigr ) \end{aligned}$$

(6.17b)

$$\begin{aligned} \sqrt{-\lambda _-} \,a_3= & {} \frac{q}{\sqrt{2}} \Bigl ( 1+\frac{1}{2}\,q-\frac{3}{16}\,{q}^{2}-{\frac{29}{32}}\,{q}^{3}-{\frac{411}{512}}\,{q}^{4}+{\frac{749}{1024}}\,{q}^{5}+{\frac{21955}{8192} }\,{q}^{6} \nonumber \\&+{\frac{33909}{16384}}\,{q}^{7} + O({q}^8) \Bigr ) \end{aligned}$$

(6.17c)

$$\begin{aligned} \sqrt{\lambda _+} \,b_1= & {} \frac{1}{\sqrt{2}} \Bigl ( 1+\frac{3}{16}\,{q}^{2}-{\frac{83}{512}}\,{q}^{4}+{\frac{3605}{ 8192}}\,{q}^{6}-\frac{1}{2}\,{q}^{7} + O({q}^8) \Bigr ) \end{aligned}$$

(6.18a)

$$\begin{aligned} \sqrt{\lambda _+} \,b_2= & {} \frac{i}{\sqrt{2}} \Bigl ( 1 +\frac{1}{2}\,q -\frac{3}{16}\,{q}^{2} +{\frac{3}{32}}\,{q}^{3} +{\frac{101}{512}}\,{q}^{4} -{\frac{595}{1024}}\,{q}^{5} +{\frac{4035}{8192}}\,{q}^{6} \nonumber \\&+{\frac{10261}{16384}}\,{q}^{7} + O({q}^8) \Bigr ) \end{aligned}$$

(6.18b)

$$\begin{aligned} \sqrt{\lambda _+} \,b_3= & {} -\frac{q}{\sqrt{2}} \Bigl ( 1-\frac{1}{2}\,q-\frac{3}{16}\,{q}^{2}+{\frac{29}{32}}\,{q}^{3}-{ \frac{411}{512}}\,{q}^{4}-{\frac{749}{1024}}\,{q}^{5}+{\frac{21955}{8192}}\,{q}^{6} \nonumber \\&-{\frac{33909}{16384}}\,{q}^{7} + O({q}^8) \Bigr ) \end{aligned}$$

(6.18c)

$$\begin{aligned} \sqrt{\lambda _0} \,c_1= & {} -{q}^{7/2} \left( 1-2\,{q}^{2}+{q}^{4}+7\,{q}^{6} + O({q}^8) \right) \end{aligned}$$

(6.19a)

$$\begin{aligned} \sqrt{\lambda _0} \,c_2= & {} i{q}^{5/2} \left( 1-{q}^{2}-{q}^{4}+7\,{q}^{6} + O({q}^8) \right) \end{aligned}$$

(6.19b)

$$\begin{aligned} \sqrt{\lambda _0} \,c_3= & {} {q}^{3/2} \left( 1-{q}^{2}+4\,{q}^{6} + O({q}^8) \right) \end{aligned}$$

(6.19c)