(frontrun buyers), then $w_{1}\to w_{1}+di$, $w_{2}\to w_{2}$. If $P^{f}$ was
traded at $p_{2}$ (frontrun sellers), then $w_{1}\to w_{1}$, $w_{2}\to
w_{2}+di$. (In both cases $p_{\\{1,2\\}}$ do not change.). These two
alternatives (frontrun buyers/frontrun sellers) give identical price change,
and, if $di\leq w_{\\{1,2\\}}$, also give identical maximal P&L. Otherwise a
term $\min(di,w_{\\{1,2\\}})$ similar to the one in (171) arise.

To obtain directional information we need a criteria to distinguish the two
alternatives. They can be distinguished considering variations of $P^{f}$.
Assume execution flow to occur not at specific single price $P^{f}$, but
within some price interval $P^{f}\pm\Delta p$. Note that according to
time–price symmetry argumentMalyshkin and Bakhramov (2015) first order
derivative cannot privide dynamics information, thus the P&L should be
invariant with respect to $\Delta p\to-\Delta p$. Consider the P&L
corresponding to impact from the future execution flow $di$, with $P^{f}$,
distributed within the $p_{\\{1,2\\}}\pm\Delta p$ interval. Then

$\displaystyle\mathrm{P\&L_{fr\,long}}(P^{f})$ $\displaystyle=$
$\displaystyle(p_{2}(P^{f})-P^{f})\min(di,w_{2})$ (172)
$\displaystyle\mathrm{P\&L_{fr\,short}}(P^{f})$ $\displaystyle=$
$\displaystyle(P^{f}-p_{1}(P^{f}))\min(di,w_{1})$ (173)
$\displaystyle\Delta\mathrm{P\&L_{fr}}$ $\displaystyle=$
$\displaystyle\mathrm{P\&L_{fr\,long}}(p_{2}\pm\Delta
p)-\mathrm{P\&L_{fr\,short}}(p_{2}\pm\Delta p)$ (174)

The $p_{\\{1,2\\}}(P^{f})$ is a function with min/max at
$P^{f}=p_{\\{2,1\\}}$, see Appendix D Fig. 14. Long/short assymetry (174), can
be considered as directional asymmetry and for infinitesimal $\Delta p$ second
order term is:

$\displaystyle\Delta\mathrm{P\&L_{fr}}$ $\displaystyle\approx$
$\displaystyle\frac{(\Delta
p)^{2}}{2}\left[\left.\frac{\partial^{2}\mathrm{P\&L_{fr\,long}}(P^{f})}{\partial(P^{f})^{2}}\right|_{P^{f}=p_{1}}-\left.\frac{\partial^{2}\mathrm{P\&L_{fr\,short}}(P^{f})}{\partial(P^{f})^{2}}\right|_{P^{f}=p_{2}}\right]$
(175) $\displaystyle=$ $\displaystyle\frac{(\Delta
p)^{2}}{2}\left[\left.\min(di,w_{2})\frac{\partial^{2}p_{2}(P^{f})}{\partial(P^{f})^{2}}\right|_{P^{f}=p_{1}}+\left.\min(di,w_{1})\frac{\partial^{2}p_{1}(P^{f})}{\partial(P^{f})^{2}}\right|_{P^{f}=p_{2}}\right]$
(176)

The $p_{\\{1,2\\}}(P^{f})$ are similar to solid state physics ‘‘band
structure’’. It is convinient to introduce an ‘‘effective mass’’ near zone
edge:

$\displaystyle\frac{1}{m_{1}}$ $\displaystyle=$
$\displaystyle\left.\frac{\partial^{2}p_{1}(P^{f})}{\partial(P^{f})^{2}}\right|_{P^{f}=p_{2}}$
(177) $\displaystyle\frac{1}{m_{2}}$ $\displaystyle=$
$\displaystyle\left.\frac{\partial^{2}p_{2}(P^{f})}{\partial(P^{f})^{2}}\right|_{P^{f}=p_{1}}$
(178) $\displaystyle{\cal D}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{1}}+\frac{1}{m_{2}}$ (179)

We have $m_{1}>0$ and $m_{2}<0$, as for electrons and holes in a
semiconductor, see Fig. 15 for this ‘‘transition’’ analogy. The ${\cal D}$,
directional assymetry of distribution, is related to distribution skewness
(74) and, in some situations, can be used as a directional indicator.

##  Appendix F  Future Wavefunction Without $I_{0}^{f}$.

In the section VII.3 we have determined (44) future $I_{0}^{f}$ and made an
attempt to convert this information to price information using the dynamic
equation of Ref. Malyshkin and Bakhramov (2015). A question arise what kind of
answer can be obtained without information about $I_{0}^{f}$? It is clear,
that in this case only perturbation theory on ${dI}/{I_{0}^{f}}$ can be
developed. Because $dI\geq 0$ (46) some information can still be obtained,
even in case of unknown $I_{0}^{f}$ value.

Figure 16:  The AAPL stock price on September, 20, 2012. Calculated in Shifted
Legendre basis with $n=7$ and $\tau$=128sec. The $P^{[IH]}$, $P_{1}$ and
$P_{2}$ are calculated according to (61), (191) and (192) respectively.

Consider some wavefunction $\psi(x)$ and corresponding execution flow
$I_{\psi}$, calculated as in (24). Consider simple variation $\delta\psi(x)$.
Then second order Rayleigh quotient perturbation is:

$\displaystyle I_{\psi+\delta\psi}$ $\displaystyle=$
$\displaystyle\frac{\Braket{\psi+\delta\psi}{I}{\psi+\delta\psi}}{\Braket{\psi+\delta\psi}{\psi+\delta\psi}}=D0+D1+D2+\dots$
(180) $\displaystyle D0$ $\displaystyle=$
$\displaystyle\frac{\Braket{\psi}{I}{\psi}}{\Braket{\psi}{\psi}}$ (181)
$\displaystyle D1$ $\displaystyle=$ $\displaystyle
2\left(\frac{\Braket{\psi}{I}{\delta\psi}}{\Braket{\psi}{\psi}}-D0\frac{\Braket{\psi}{\delta\psi}}{\Braket{\psi}{\psi}}\right)$
(182) $\displaystyle D2$ $\displaystyle=$
$\displaystyle\frac{\Braket{\delta\psi}{I}{\delta\psi}}{\Braket{\psi}{\psi}}-D0\frac{\Braket{\delta\psi}{\delta\psi}}{\Braket{\psi}{\psi}}-2\frac{\Braket{\psi}{\delta\psi}}{\Braket{\psi}{\psi}}D1$
(183)

A rather complex perturbation theory on $\Ket{\delta\psi}$ can be developed in
a style of our earlier work Malyshkin  _et al._ (1999) in a very different
field of multiple-scattering, but we limit here all the considerations to
first order $I$ variation only on $\delta\psi$ states, orthogonal to $\psi$,
i.e. $\Braket{\delta\psi}{\psi}=0$. Then

$\displaystyle I_{\psi+\delta\psi}$ $\displaystyle\approx$
$\displaystyle\frac{\Braket{\psi}{I}{\psi}}{\Braket{\psi}{\psi}}+\delta I$
(184) $\displaystyle\delta I$ $\displaystyle=$ $\displaystyle
2\frac{\Braket{\psi}{I}{\delta\psi}}{\Braket{\psi}{\psi}}=2\Braket{b}{\delta\psi}$
(185) $\displaystyle\Ket{b}$ $\displaystyle=$ $\displaystyle\Ket{I|\psi}$
(186)

Thus $\delta I$ (185) is represented as a scalar product of $\Ket{b}$ and
$\Ket{\delta\psi}$ vectors. What variation $\delta\psi$ to provide maximal
$\delta I$? The one, different from $\Ket{b}$ only on a constant $\beta$, i.e.

$\displaystyle\Ket{\phi}$ $\displaystyle=$
$\displaystyle\Ket{I|\psi}-\Braket{\psi}{I}{\psi}\Ket{\psi}$ (187)
$\displaystyle\Ket{\delta\psi}$ $\displaystyle=$
$\displaystyle\Ket{\phi}\beta$ (188)

The states (188) provide maximal variation $\delta I$. The term
$\Braket{\psi}{I}{\psi}\Ket{\psi}$ is subtracted in (187) to have
$\Braket{\delta\psi}{\psi}=0$. Put $\Ket{\psi}=\Ket{\psi_{0}}$ from Eq. (39)
to (187), this immediately lead to $\phi(x_{0})=0$, and consider $I$ as a
function of $\beta$

$\displaystyle\Ket{\phi}$ $\displaystyle=$
$\displaystyle\Ket{I|\psi_{0}}-\Braket{\psi_{0}}{I}{\psi_{0}}\Ket{\psi_{0}}$
(189) $\displaystyle I(\beta)$ $\displaystyle=$
$\displaystyle\frac{\Braket{\psi_{0}+\beta\phi}{I}{\psi_{0}+\beta\phi}}{\Braket{\psi_{0}+\beta\phi}{\psi_{0}+\beta\phi}}$
(190) $\displaystyle\approx$ $\displaystyle
I_{0}+2\beta\Braket{\phi}{I}{\psi_{0}}+\dots$

As we noted in section VII.1 when $\Ket{\psi_{0}}$ is an eigenfunction of (33)
(or $I=const$ and the problem (33) is degenerated), then theory fails (now for
the reason of $\Braket{\phi}{\phi}=0$ no first order perturbation theory
possible). Otherwise, because
$\Braket{\phi}{I}{\psi_{0}}=\Braket{\phi}{\phi}>0$ we always have $\beta>0$
and in the first order perturbation two answers, let us call them, $P_{1}$ and
$P_{2}$ in a weak hope to get a poor–man $P^{[IH]}$:

$\displaystyle P_{1}$ $\displaystyle=$
$\displaystyle\frac{\Braket{\phi}{pI}{\psi_{0}}}{\Braket{\phi}{I}{\psi_{0}}}$
(191) $\displaystyle P_{2}$ $\displaystyle=$
$\displaystyle\frac{\Braket{\phi}{pI}{\phi}}{\Braket{\phi}{I}{\phi}}$ (192)
$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\Braket{\phi}{\phi}}}{\Braket{\psi_{0}}{I}{\psi_{0}}}=\frac{\sqrt{\Braket{\psi_{0}}{I}{I|\psi_{0}}-\Braket{\psi_{0}}{I}{\psi_{0}}^{2}}}{\Braket{\psi_{0}}{I}{\psi_{0}}}$
(193)

These answers, while being very crude estimates in practice, may be still
useful (especially $P_{2}$ from (192)) in applications for their simplicity.
The $r$ (standard deviation –like estimate of $I$ on $\Ket{\psi_{0}}$ state)
from (193) can serve as an estimate of how close is $\Ket{\psi_{0}}$ to
$\|I\|$ eigenfunction. The major drawback of all these first order
perturbation answers is that they are not as good in automatic selection of
proper time–scale, as eigenvalues problem. In Fig. 16 the $P^{[IH]}$, $P_{1}$
and $P_{2}$ are presented (calculated according to (61), (191) and (192)
respectively). One can see that the $P_{2}$ has a similar to $P^{[IH]}$
behavior, especially it tracks well market direction change. The $P_{1}$,
because it is not averaged with always positive weight, is more volatile than
$P_{2}$, but also can be of interest. A very important feature of $P_{1}$
(191) and $P_{2}$ (192) is that they are obtained without solving eigenvalues
problem, but, nevertheless, still provide some information, thus can be
considered as a poor man $P^{[IH]}$.

##  Appendix G  Computer Code Implementation

###  G.1  Installation and Data Preparation

  * •

Install java 1.8 or later.

  * •

Download from Malyshkin (2014) NASDAQ ITCH data file S092012-v41.txt.gz, and
the archive AMuseOfCashFlowAndLiquidityDeficit.zip with the source code.

  * •

Decompress and recompile the program:

        unzip AMuseOfCashFlowAndLiquidityDeficit.zip
    javac -g com/polytechnik/*/*java

  * •

Extract triples (time, execution price, shares traded) from NASDAQ ITCH data
file:

        java com/polytechnik/itch/DumpData2Trader \
          S092012-v41.txt.gz AAPL >aapl.csv

Execution data and limit order book edges are now saved to tab–separated file
`aapl.csv` of 15 columns and 634205 lines. The columns of interest are:

    * –

`currenttime` Time in nanoseconds since midnight.

    * –

`exe_price_last` Last Price.

    * –

`exe_shares` Shares traded.

  * •

Run the command to test the program

        java com/polytechnik/algorithms/CallAMuseOfCashFlowAndLiquidityDeficit \
          --musein_cols=15:1:4:5 \
          --musein_file=aapl.csv \
          --museout_file=museout.dat \
          --n=7 \
          --tau=128 \
          --measure=ImpactQVMMuseLegendreShifted

Program parameters are:

    * `--musein_file=aapl.csv` : Input tab–separated file with (time, execution price, shares traded) triples timeserie.

    * `--musein_cols=15:1:4:5` : Out of total 15 columns of `aapl.csv` file, take column #1 as time (nanoseconds since midnight), #4 (execution price), and #5 (shares traded), column index is base 0.

    * `--museout_file=museout.dat` : Output file name is set to museout.dat.

    * `--n=7` : Basis dimension. Typical values are: 2 (for testing a concept), or some value about $[4\dots 12]$ for more advance use.

    * `--tau=128` : Exponent time (in seconds) for the measure used.

    * `--measure=ImpactQVMMuseLegendreShifted` The measure. The values ImpactQVMMuseLaguerre,ImpactQVMMuseLegendreShifted,ImpactQVMMuse_pi correspond the measures (7), (14), (19) respectively. The results of ImpactQVMMuseMonomials (uses $Q_{k}(x)=x^{k}$) should be identical to ImpactQVMMuseLaguerre (uses $Q_{k}(x)=L_{k}(-x)$), as the measure is the same and all the calculations are $Q_{k}(x)$–basis invariant (but numerical stability is worse for ImpactQVMMuseMonomials).

  * •

The results are saved in the output file `museout.dat`.

  * •

There is a short ‘‘bundled’’ data file dataexamples/aapl_old.csv.gz of 9
colums and 28492 lines, that contains only executions (no limit order book
events). It can be used for testing insead of aapl.csv obtained from
S092012-v41.txt.gz:

        java com/polytechnik/algorithms/CallAMuseOfCashFlowAndLiquidityDeficit \
          --musein_cols=9:1:2:3 \
          --musein_file=dataexamples/aapl_old.csv.gz
          --museout_file=museout.dat \
          --n=7 \
          --tau=128 \
          --measure=ImpactQVMMuseLegendreShifted

###  G.2  CallAMuseOfCashFlowAndLiquidityDeficit.java

Output file is tab–separated file with columns corresponding the calculations
of this paper. Most output data is saved in the objects of Skewness type
(skewness and generalized skewness) and EVXData type (generalized eigenvalue
problem $\Ket{I|\psi}=\lambda\Ket{\psi}$) created by the ImpactQVMMuse. Field
number (and name) are printed in the first line of output file, so they can be
processed by any common plotting software (such as gnuplot or matlab). Below
are the description of most noticeable fields:

  * •

`T` Time in nanoseconds since midnight (copied from input).

  * •

`shares` Shares traded (copied from input).

  * •

`P_last` Execution price (copied from input).

  * •

`I.*` Correspond to $\Ket{I|\psi}=\lambda\Ket{\psi}$ eigenvalues solution with
the given `--n=`. The `I.Gamma0` is $\widetilde{\Gamma^{0}}$ (103) of past
sample. The `I.sL`, `I.sH`, and `I.s0` correspond to min/max eigenvalues, and
$\Braket{\psi_{0}}{I}{\psi_{0}}$. The I.wL and I.wH are squared in the output.

  * •

`P.*` Correspond to $\Ket{pI|\psi}=\lambda\Ket{I|\psi}$ eigenvalues solution
with the given `--n=`. This eigenproblem for price is presented just for
completeness.

  * •

`SK_P_IH.*` Skewness on $\max I$ state from Section IX.2, with $dI=0$. The
`SK_P_IH.xa` is equal to $P^{[IH]}$ (61).

  * •

`pnlss.*` fields correspond to $n=2$ (regardless of the given `--n=` value,
use Laguerre basis to have $p^{k}$ and $t^{k}$ basis similar behavior without
\--tau= adjustment), calculations of Section X. Regular price skewness (74)
along with the generalized skewness (100) for $I$ and $P$ are presented.
Regular exponential moving average
$\overline{p}_{\tau}=\Braket{Q_{0}pI}/\Braket{Q_{0}I}$ is equal to any of
pnlss.{SK_P_average,gSK_P_average}.xa, and $p_{\\{1,2\\}}$ nodes (72) are
`pnlss.SK_P_average.{x1,x2}`.

  * •

pnldidsk.* fields calculated by the PnLdIDSk class, most noticeable are:
pnldidsk.SK_spur__nodI the skewness of Section IX.5 density matrix states, the
pnldidsk.SK_spur__nodI.xa is
$\overline{p}^{spur}_{\tau}=Spur(\|pI\|)/Spur(\|I\|)$, moving average,
calculated via operator spur (sum of diagonal elements). The
pnldidsk.Pf_from_pt_true_pi is (60).

  * •

pnlfutureSk.* correspond to Section X.3 calculations.

Current CallAMuseOfCashFlowAndLiquidityDeficit.java output 77 fields. The code
can be modified to adjust the output. You may also use
com/polytechnik/scripts/plot_chart.pl to select only specific fields, also you
may run com/polytechnik/trading/GenerateTrainingData.java to produce more data
in output.

###  G.3  Code Structure

The codebase is huge. Most of the code are my past fault attempts to find a
market dynamics equation. Once an idea is decided to be a fault — all related
code is moved to the unit tests, thus increase the codebase444 This section is
adjusted from the earlier version in order to reflect API changes in Malyshkin
(2019). . To run all unit tests at once execute the command:

    java com/polytechnik/trading/QVM

It may take a while to finish all the unit tests (about 2 days to run, the
best usage I found for these unit tests is to catch Java HotSpot JIT compiler
bugs $\stackrel{{\scriptstyle\cdot\cdot}}{{\frown}}$). But for the theory of
this paper the calculations are extremely fast and there are actually very few
classes of interest. Most noticeable of them are described below.

    com/polytechnik/trading/QVMDataL.java
    com/polytechnik/trading/QVMDataP.java
    com/polytechnik/trading/QVMData.java

These calculate the moments $\Braket{fQ_{k}}$ from a sequence of trades using
Laguerre, Shifted Legendre, or monomials basis for $Q_{k}(x)$. The
calculations are optimized to incrementally555Using the
$Q_{n}(ax+b)=\sum_{k=0}^{n}d^{(n)}_{k}Q_{k}(x)$ expansion, that is Newton
Binomial $(1+x)^{n}=\sum_{k=0}^{n}C_{n}^{k}x^{k}$ monomials basis
generalization. For numerical implementation see setNewtonBinomialLikeCoefs
method of classes extending the com/polytechnik/utils/BasisPolynomials.java
class, implementing the expansion using three term recurrence of basis
polynomials $Q_{k}(x)$, see Appendix A “Non-monomials polynomial bases” of
Ref. Malyshkin and Bakhramov (2015). update already calculated moments, what
make the calculations extremely fast, thus applicable to a practical realtime
HFT trading. To access calculated distribution moments use the classes
implementing the DataInterfaceToMoments<T>:

    com/polytechnik/trading/QVMDataLDirectAccess.java
    com/polytechnik/trading/QVMDataPDirectAccess.java
    com/polytechnik/trading/QVMDataDirectAccess.java

To manipulate distribution moments obtained in various $Q_{k}(x)$ bases there
are few classes (they all extend the `OrthogonalPolynomialsABasis` and use a
reference to `BasisPolynomials` to manipulate polynomials):

    com/polytechnik/utils/OrthogonalPolynomialsLegendreShiftedBasis.java
    com/polytechnik/utils/OrthogonalPolynomialsLegendreBasis.java
    com/polytechnik/utils/OrthogonalPolynomialsLaguerreBasis.java
    com/polytechnik/utils/OrthogonalPolynomialsChebyshevBasis.java
    com/polytechnik/utils/OrthogonalPolynomialsHermiteEBasis.java
    com/polytechnik/utils/OrthogonalPolynomialsMonomialsBasis.java

Once the moments $\Braket{fQ_{k}}$ are calculated from a sequence of trades,
the classes such as:

    com/polytechnik/trading/MomentsData.java
    com/polytechnik/trading/SMomentsData.java

calculate and store the matrices: $\Braket{Q_{j}}{Q_{k}}$,
$\Braket{Q_{j}}{I}{Q_{k}}$, $\Braket{Q_{j}}{pI}{Q_{k}}$,
$\Braket{Q_{j}}{dp/dt}{Q_{k}}$ (and others, the classes are different in
attributes selection) from the moments data using basis functions
multiplication operator $c_{l}^{jk}$ :

$\displaystyle Q_{j}(x)Q_{k}(x)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{j+k}c_{l}^{jk}Q_{l}(x)$ (194)

The $c_{l}^{jk}$ coefficients are available analytically for all practically
interesting bases, see Appendix A of Ref. Malyshkin and Bakhramov (2015) and
references therein, the calculations are implemented in the classes above, the
ones extending the OrthogonalPolynomialsABasis and BasisPolynomials. The
class:

    com/polytechnik/utils/EVXData.java

given two matrices $\Braket{Q_{j}}{Q_{k}}$ and $\Braket{Q_{j}}{I}{Q_{k}}$
solves generalized eigenvalue problem
$\Ket{I|\psi_{I}^{[i]}}=\lambda_{I}^{[i]}\Ket{\psi_{I}^{[i]}}$, finds
eigenvalues and eigenvectors, calculates $\Braket{\psi_{0}}{\psi_{I}^{[i]}}$
projections, and $I_{0}=\Braket{\psi_{0}}{I}{\psi_{0}}$. The class:

    com/polytechnik/trading/PnLSimpleSkewness.java

perform simple calculations of Sections IX.1, X.1, and X.2 (for $n=2$ all the
matrices are $2\times 2$). This class calculates: price regular skewness
(skewness, quadrature nodes, and weights are calculated), generalized skewness
($\widetilde{\Gamma}$ skewness, $\widetilde{\Gamma^{0}}$ skewness,
$\lambda^{[\\{0,1\\}]}$, and weights), and, out of curiosity, probability
correlation $\widetilde{\rho}(p,I)$ of Appendix C. The class:

    com/polytechnik/trading/PnLdIDSk.java

performs naïve dynamic impact calculations of Section VII.4 along with some
other skewness–related calculations considered in Sections IX.2 and IX.5. The
class

    com/polytechnik/trading/PnLFutureSk.java

performs the calculations of Section X.3. It takes an instance of
`MomentsData` and do the following:

  * •

Solve generalized eigenvalue problem (33), find $dI$ as (45) and $P^{[IH]}$ as
(61).

  * •

Construct $\|I^{f}\|$ operator (51).

  * •

Solve generalized eigenvalue problem (52).

  * •

Find past $\widetilde{\Gamma^{0}}$ and future $\widetilde{\Gamma^{0}}^{f}$
skewness of $I$.

  * •

Find $P^{f}$ as (106).

This class demonstrate reference implementation of this paper theory:

    com/polytechnik/algorithms/CallAMuseOfCashFlowAndLiquidityDeficit.java

It read line-by-line tab–separated timeserie file of triples (time, execution
price, shares traded) to update a sequence of executed trades. For each new
trade (new line read), it calls666 See the file
com/polytechnik/ImpactQVMMuse.java, that, create MomentsData out of trade
sequence, do the calculations, and save the results to (type:name) objects:
PnLSimpleSkewness:pnlss, Skewness:SK_IH, PnLdIDSk:pnldidsk,
PnLFutureSk:pnlfutureSk, and along with (for demonstration) separate
calculation of $P^{[IH]}$ from (61) and $w^{[IL]}_{I}$, $w^{[IH]}_{I}$
projections (42), (43) (using EVXData:I).  com/polytechnik/ImpactQVMMuse<T>,
that incrementally (optimization for speed) calculates the moments, obtains
the MomentsData with $\Braket{Q_{j}}{Q_{k}}$, $\Braket{Q_{j}}{I}{Q_{k}}$,
$\Braket{Q_{j}}{pI}{Q_{k}}$ matrices, performs the calculations and creates
the ImpactQVMMuse object, then, finally, outputs the data out of the
ImpactQVMMuse as described in the previous section.

## References

  * Malyshkin and Bakhramov (2015) Vladislav Gennadievich Malyshkin and Ray Bakhramov, ‘‘Mathematical Foundations of Realtime Equity Trading. Liquidity Deficit and Market Dynamics. Automated Trading Machines.’’ ArXiv e-prints (2015), http://arxiv.org/abs/1510.05510, arXiv:1510.05510 [q-fin.CP] .
  * Malyshkin (2016) Vladislav Gennadievich Malyshkin, ‘‘Market Dynamics. On Supply and Demand Concepts,’’ ArXiv e-prints (2016), http://arxiv.org/abs/1602.04423, arXiv:1602.04423 .
  * Malyshkin and Bakhramov (2016) Vladislav Gennadievich Malyshkin and Ray Bakhramov, ‘‘Market Dynamics vs. Statistics: Limit Order Book Example,’’ ArXiv e-prints (2016), arXiv:1603.05313 [q-fin.TR] .
  * Hautsch and Huang (2011) Nikolaus Hautsch and Ruihong Huang, ‘‘Limit order flow, market impact and optimal order sizes: Evidence from nasdaq totalview-itch data,’’  (2011).
  * Wikipedia (2016a) Wikipedia, ‘‘Market impact,’’  (2016a).
  * Malyshkin (2019) Vladislav Gennadievich Malyshkin, ‘‘Market Dynamics: On Directional Information Derived From (Time, Execution Price, Shares Traded) Transaction Sequences.’’ ArXiv e-prints (2019), arXiv:1903.11530 [q-fin.TR] .
  * Boiko and Malyshkin (27 Feb. 2009) Andrei Mikhailovich Boiko and Vladislav Gennadievich Malyshkin, ‘‘International Winter School on Physics of Semiconductors, Zelenogorsk,’’ Personal communication (27 Feb. 2009), we have had a philosophical discussion on experimental proof of time machine existence. SciFi models of physical time travel on any time distance is a pipe dream, assume you have some real time machine (of dynamic equation type), that does not allow a physical time travel, but allows to look ahead a few seconds, (or even a fraction of a second) and then report back information from the future. How to prove that such a time machine actually works? The best answer we found is this: Attach this time machine to an exchange and show the P&L! If you cornered the market — it works. Another interesting effect of time machine existence would be a destabilization of the financial markets when used on mass scale. Any arbitrage–type trading strategy stop working when started to be applied by multiple market participants (a special case of Goodhart’s lawWikipedia (2016e) Any observed statistical regularity will tend to collapse once pressure is placed upon it for control purposes). In contrast, a trading strategy that deploys the time machine does not stop working when applied by multiple market participants: real time machine plain adjusts the prediction based on new market participants coming. The strategy is now confined not by the quality of future prediction, but by the inability to perform an action, e.g. inability to execute a trade "now". (If this inability is caused by the liquidity deficit, then a severe price move is expected). A complete inability to execute a transaction (due to whatever reason) means market disappearance. Important, that other known systems with a dynamic equation (e.g. Newton Mechanics, Maxwell Electrodynamics, Schrödinger equation in Quantum Mechanics) while have the future accurately predicted, do not have the described feedback loop of the financial markets, when a prediction from the dynamic equation change the behavior of market participants, thus change the future and cause destabilization. Same conclusion can be also obtained from a "religious interpretation" of a trading system. For an adept of quite popular these days "religion of money", the P&L of a trading system (any system, not necessary an automatic trading machine), is the only criteria, separating true and false prophets apart. Moreover, generated P&L of a trading systems, run by the True Prophet, should not fade, when other market participants start deploying the same strategy. What will happen to the markets when the True Prophet of the "religion of money" come to Earth? Same thing, as if the Dynamic Equation exists and can be solved: the markets now to be confined not by arbitrage strategy fading, but by inability to execute a transaction, i.e. by the Liquidity Deficit.
  * Hawkins and Blakeslee (2007) Jeff Hawkins and Sandra Blakeslee,  _On Intelligence_ (Macmillan, 2007).
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  * Malyshkin (2015a) Vladislav Gennadievich Malyshkin, ‘‘Radon–Nikodym approximation in application to image analysis.’’ ArXiv e-prints (2015a), http://arxiv.org/abs/1511.01887, arXiv:1511.01887 [cs.CV] .
  * Wikipedia (2016b) Wikipedia, ‘‘Bra–ket notation,’’  (2016b).
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# Magnetic Neumann Laplacian on a domain with hole

Diana Barseghyan1,2111corresponding author, Swanhild Bernstein3 and Baruch
Schneider1

(1 Department of Mathematics, University of Ostrava, 30.dubna 22, Ostrava
70103, Czech Republic
2 Department of Theoretical Physics
Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež, Czech
Republic
3 Institute of Applied Analysis, TU Bergakademie Freiberg, Akademiestrasse 6,
Freiberg 09599, Germany
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>)

###### Abstract

This article gives a domain with a small compact set of removed and the
magnetic Neumann Laplacian on such set. The main theorem of this article shows
the description of the holes which do not change the spectrum drastically. In
this article we prove that the spectrum of the magnetic Neumann Laplacian
converges in the Hausdorff distance sense to the spectrum of the original
operator defined on the unperturbed domain.

Keywords. Magnetic Neumann Laplacian, spectral convergence, domain with hole.

Mathematics Subject Classification. Primary: 58J50; Secondary: 35P15, 47A10.

##  1 Introduction

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded domain and a compact set
$K\subset\Omega$. We denote $\Omega_{K}:=\Omega\setminus K$. The magnetic
Neumann Laplacian

$H_{\Omega_{K}}(A_{\mathrm{magn}})=(i\nabla+A_{\mathrm{magn}})^{2}$

is defined via the quadratic form

$\int_{\Omega_{K}}|i\nabla u+A_{\mathrm{magn}}u|^{2}dxdy,\quad
u\in\mathcal{H}^{1}(\Omega_{K}),$

where $A_{\mathrm{magn}}$ is a magnetic vector potential generating the
magnetic field

$\mathrm{rot}(A_{\mathrm{magn}})(x,y)=B(x,y).$

However for applications, a better do not use different notation for the
magnetic potentials corresponding to the restrictions of the same magnetic
field. The first motivation arises for studying the magnetic Laplacian from
the mathematical theory of superconductivity, see e.g [10]. The second
motivation is to understand at which point there is an analogy between the a
magnetic Laplacian and a non-magnetic Laplacian. In this paper we will focus
on problems with magnetic Laplacian.

It is a common expectation that small perturbations of the physical situation
lead only to a small change of the spectrum. In the case of domain
perturbations this is largely true for Dirichlet boundary conditions while the
Neumann case is more delicate.

In Neumann case even small perturbations may cause abrupt change of the
spectrum. For example, such an effect is observed when the hole has a ”split-
ring” geometry [11]. The split ring (even being very small) may produce
additional eigenvalues having nothing in common with the eigenvalues of the
Neumann Laplacian on the unperturbed domain.

The problems with the Neumann obstacles having more general geometry appeared
in [9]. The authors required the hole to satisfy the so called ”uniform
extension property”. This is a requirement, which means that
$\mathcal{H}^{1}$\- functions on the domain with a hole can be extended to
$\mathcal{H}^{1}$ function on the unperturbed domain and the norm of this
extension operator does not depend on the hole diameter. In this case the
authors established the spectral convergence.

In [1] it was considered the spectral problem for the Neumann Laplacian on the
domain with hole with zero Lebesgue measure. Obviously such domain does not
satisfy the uniform extension property but under some additional assumption on
hole one is able to prove the spectral convergence.

In the present work we discuss the similar situation where one is not able to
guarantee the validity of the uniform extension property. Here we do not
restrict ourselves that the hole must have zero Lebesgue measure. Moreover, we
extend the class of operators: instead of the Laplacian we consider the
magnetic Laplace operators.

Before the description of our results let as also mention paper [8], which
makes use imposing Neumann boundary conditions on the boundary of hole but in
higher dimension. The situation with a smooth hole is considered in [3].

The second section of the present work is devoted to the description of the
geometry of holes which do not change the spectrum drastically. More
precisely, we will be interested in the spectral properties of the magnetic
Neumann Laplacian defined on a two-dimensional bounded domain with a single
hole $K_{\varepsilon}$ (for a fixed parameter $\varepsilon$) having ”standard”
geometry. In this section we present the results about the spectral
convergence of the magnetic Neumann Laplacian on $\Omega_{K_{\varepsilon}}$ as
$\varepsilon\to 0$ in terms of the Hausdorff distance under some additional
assumptions on the geometry of $K_{\varepsilon}$. In the third section we
present the main tools of the spectral convergence of operators on varying
Hilbert spaces. In the fourth and fifth sections we prove our results.

##  2 Main results

Since we are working with the geometry of domains, we introduce the additional
property∗: for any $(x_{0},y_{0})\in\Omega_{K}$ at least one of the following
conditions takes place

  * •

Let $l(x_{0})=\\{x=x_{0}\\}$. Then at least one of the half-lines
$l(x_{0})\cap\\{y\geq y_{0}\\}$ and $l(x_{0})\cap\\{y\leq y_{0}\\}$ has no
intersection with $K$.

  * •

Let $h(y_{0})=\\{y=y_{0}\\}$. Then at least one of the half-lines
$h(y_{0})\cap\\{x\geq x_{0}\\}$ and $h(y_{0})\cap\\{x\leq x_{0}\\}$ has no
intersection with $K$.

Figure 1: Example of $\Omega_{K}$ not satisfying property∗

We collect in the following theorem some important results.

######  Theorem 2.1.

Let $\Omega$ be an open bounded domain in $\mathbb{R}^{2}$ and let
$p\in\Omega$ be some fixed point. Suppose that
${\mathbb{B}}_{\varepsilon}\subset\Omega$ is a ball with center at $p$ and
radius $\varepsilon>0$. Let $K=K(\varepsilon)\subset\mathbb{B}_{\varepsilon}$
be a compact set. Moreover, suppose that $\Omega_{K}$ satisfies property∗. Let
$H_{\Omega}(A_{\mathrm{magn}})$ and
$H_{\Omega_{K_{\varepsilon}}}(A_{\mathrm{magn}})$ be the magnetic Neumann
Laplacians defined on $\Omega$ and $\Omega_{K_{\varepsilon}}$, respectively.
We assume that on some subset of $\Omega$ containing
${\mathbb{B}}_{\varepsilon}$ the vector potential $A_{\mathrm{magn}}$ is twice
continiously differentiable. Then for small enough $\varepsilon$ there exists
$\eta(\varepsilon)>0$ with $\eta(\varepsilon)\to 0$ as $\varepsilon\to 0$ such
that the following spectral convergence takes place

$\overline{d}\left(\sigma_{\bullet}\left(H_{\Omega_{K_{\varepsilon}}}(A_{\mathrm{magn}})\right),\,\sigma_{\bullet}\left(H_{\Omega}(A_{\mathrm{magn}})\right)\right)\leq\eta(\varepsilon),$

where $\overline{d}$ is defined in (3.9) and $\sigma_{\bullet}(\cdot)$ denotes
either the entire spectrum, the essential or the discrete spectrum. Moreover,
the multiplicity of the discrete spectrum is preserved.

∎

The previous result motivates the following consequences:

######  Corollary 2.2.

Suppose that $H_{\Omega}(A_{\mathrm{magn}})$ has purely discrete spectrum
denoted by $\lambda_{k}(\Omega)$ (repeated according to multiplicity). Then
the infimum of the essential spectrum of
$H_{\Omega_{K_{\varepsilon}}}(A_{\mathrm{magn}})$ tends to infinity and there
exists $\eta_{k}(\varepsilon)>0$ with $\eta_{k}(\varepsilon)\to 0$ as
$\varepsilon\to 0$ such that

$|\lambda_{k}(\Omega)-\lambda_{k}(\Omega_{K_{\varepsilon}})|\leq\eta_{k}(\varepsilon)$

for small enough $\varepsilon$. Here, $\lambda_{k}(\Omega_{K_{\varepsilon}})$
denotes the discrete spectrum of
$H_{\Omega_{K_{\varepsilon}}}(A_{\mathrm{magn}})$ (below the essential
spectrum) repeated according to multiplicity.

######  Corollary 2.3.

The Hausdorff distance between the spectra of
$H_{\Omega_{K_{\varepsilon}}}(A_{\mathrm{magn}})$ and
$H_{\Omega}(A_{\mathrm{magn}})$ converges to zero on any compact interval
$[0,\Lambda]$.

The proof of Theorem 2.1 is based on Theorem 3.1 and on the following theorem:

######  Theorem 2.4.

Under the assumptions of Theorem 2.1 for small enough $\varepsilon$ the
operators $H_{\Omega}(A_{\mathrm{magn}})$ and
$H_{\Omega_{K_{\varepsilon}}}(A_{\mathrm{magn}})$ are
$\mathcal{O}\left(\varepsilon^{1/6}\right)$ close of order $2$.

##  3 Main tool of the spectral convergence of operators on varying Hilbert
spaces

We begin this section by reviewing some basic facts which assures a spectral
convergence for two operators having the different domains. For more
information we refer the reader to [7].

To a Hilbert space $H$ with inner product $(\cdot,\cdot)$ and norm $\|\cdot\|$
together with a non-negative, unbounded, operator $A$, we associate the scale
of Hilbert spaces

$H_{k}:=\mathrm{Dom}((A+1)^{k/2}),\quad\|u\|_{k}:=\|(A+1)^{k/2}u\|,\,\,k\geq
0,$

where $1$ is the identity operator.

We think of $(H^{\prime},A^{\prime})$ being some perturbation of $(H,A)$ and
want to lessen the assumption such that the spectral properties are not the
same but still are close.

######  Definition 1.

(see [7]) Suppose we have linear operators

$\displaystyle J:H\longrightarrow H^{\prime},\quad\quad
J_{1}:H_{1}\longrightarrow H_{1}^{\prime}$ $\displaystyle
J^{\prime}:H^{\prime}\longrightarrow H,\quad\quad
J_{1}^{\prime}:H_{1}^{\prime}\longrightarrow H_{1}.$

Let $\delta>0$ and $k\geq 1$. We say that $(H,A)$ and
$(H^{\prime},A^{\prime})$ are $\delta$-close of order $k$ iff the following
conditions are fulfilled:

$\displaystyle\|Jf-J_{1}f\|_{0}\leq\delta\|f\|_{1},$ (3.1)
$\displaystyle|(Jf,u)-(f,J^{\prime}u)|\leq\delta\|f\|_{0}\|u\|_{0},$ (3.2)
$\displaystyle\|u-JJ^{\prime}u\|_{0}\leq\delta\|u\|_{1},$ (3.3)
$\displaystyle\quad\|Jf\|_{0}\leq 2\|f\|_{0},\,\,\,\|J^{\prime}u\|_{0}\leq
2\|u\|_{0},$ (3.4) $\displaystyle\|(f-J^{\prime}Jf)\|_{0}\leq\delta\|f\|_{1},$
(3.5) $\displaystyle\|J^{\prime}u-J_{1}^{\prime}u\|_{0}\leq\delta\|u\|_{1}$
(3.6)
$\displaystyle|a(f,J_{1}^{\prime}u)-a^{\prime}(J_{1}f,u)|\leq\delta\|f\|_{k}\|u\|_{1},$
(3.7)

for all $f,u$ in the appropriate spaces. Here, $a$ and $a^{\prime}$ denote the
sesquilinear forms associated to $A$ and $A^{\prime}$.

We denote by $d_{\mathrm{Haussdorff}}(A,B)$ the Hausdorff distance for subsets
$A,B\subset\mathbb{R}$

$d_{\mathrm{Haussdorff}}(A,B):=\mathrm{max}\left\\{\underset{a\in
A}{\mathrm{sup}}\,d(a,B),\,\underset{b\in B}{\mathrm{sup}}\,d(b,A)\right\\},$
(3.8)

where $d(a,B):=\mathrm{inf}_{b\in B}|a-b|$. We set

$\overline{d}(A,B):=d_{\mathrm{Hausdorff}}\left((A+1)^{-1},(B+1)^{-1}\right)$
(3.9)

for closed subsets of $[0,\infty)$. For the next result, which originates with
the work of O. Post [7] we have the following spectral convergence theorem in
terms of the distance $\overline{d}$.

######  Theorem 3.1.

[7] There exists $\eta(\delta)>0$ with $\eta(\delta)\to 0$ as $\delta\to 0$
such that

$\overline{d}(\sigma_{\bullet}(A),\,\sigma_{\bullet}(A^{\prime}))\leq\eta(\delta)$
(3.10)

for all pairs of non-negative operators and Hilbert spaces $(H,A)$ and
$(H^{\prime},A^{\prime})$ which are $\delta$-close. Here,
$\sigma_{\bullet}(A)$ denotes either the entire spectrum, the essential or the
discrete spectrum of $A$. Furthermore, the multiplicity of the discrete
spectrum, $\sigma_{\mathrm{disc}}$, is preserved, i.e. if
$\lambda\in\sigma_{\mathrm{disc}}(A)$ has multiplicity $\mu>0$, then there
exist $\mu$ eigenvalues (not necessarily all distinct) of operator
$A^{\prime}$ belonging to interval
$(\lambda-\eta(\delta),\lambda+\eta(\delta))$.

∎

We now turn to the proof of Theorem 2.4.

##  4 Proof of Theorem 2.4

We split the proof of this statement into number of steps.

Step (i). Construction of the mappings $J,J^{\prime},J_{1},J_{1}^{\prime}$.

It is easy to notice that
$H=L^{2}(\Omega),\,H^{\prime}=L^{2}(\Omega_{K_{\varepsilon}}),\,A=A^{\prime}=(i\nabla+A_{\mathrm{magn}})^{2}$;
$H_{1},\,H_{1}^{\prime}$ correspond to Sobolev spaces
$\mathcal{H}^{1}(\Omega)$ and $\mathcal{H}^{1}(\Omega_{K_{\varepsilon}})$ and
$H_{2}=\mathrm{Dom}(H_{\Omega}(A_{\mathrm{magn}}))$. The norm $\|\cdot\|_{0}$
corresponds with the $L^{2}$ norm and

$\|u\|_{1}=(\|u\|^{2}_{0}+\|i\nabla
u+A_{\mathrm{magn}}u\|_{0}^{2})^{1/2},\quad\|f\|_{2}=(\|(i\nabla+A_{\mathrm{magn}})^{2}f+f\|_{0}^{2})^{1/2}.$

We define $Ju=J_{1}u=u|_{\Omega_{K_{\varepsilon}}}$ for all $u\in H$ and
$J^{\prime}u=u\chi_{\Omega_{K_{\varepsilon}}}$ for all $u\in H^{\prime}$.

Let us now construct the mapping $J_{1}^{\prime}:H_{1}^{\prime}\to H_{1}$.
Without loss of generality, assume that the ball $\mathbb{B}_{\varepsilon}$
mentioned in Theorem 2.1 and Theorem 2.4 is centered at the origin. Let
$\epsilon\in(\varepsilon,2\varepsilon)$ be a number to be chosen later and let
$\mathbb{B}_{\epsilon}\supset\mathbb{B}_{\varepsilon}$ be the ball with center
again at the origin and radius $\epsilon$,
$\Omega_{\epsilon}:=\Omega\backslash\mathbb{B}_{\epsilon}$.

We are going to construct mapping $J_{1}^{\prime}$ first for smooth functions.
For any $v\in C^{\infty}(\Omega_{K_{\varepsilon}})$ we define

$J_{1}^{\prime}v:=\begin{cases}v,\quad\text{on}\quad\Omega_{\epsilon},\\\
\frac{r}{\epsilon}\tilde{v}(\epsilon,\varphi),\quad\text{on}\quad\mathbb{B}_{\epsilon},\end{cases}$

where $\tilde{v}(r,\varphi)=v(r\cos\varphi,r\sin\varphi)$.

Now let us construct the mapping $J_{1}^{\prime}u$ for any $u\in
H_{1}^{\prime}$. Employing the approximation method described in [4, Thm.2,
5.3.2], for the fixed sequence $\\{\eta_{k}\\}_{k=1}^{\infty}$ converging to
zero we construct the sequence $v_{\eta_{k}}\in
C^{\infty}(\Omega_{K_{\varepsilon}})$ which satisfies

$\|u-v_{\eta_{k}}\|_{1}<\eta_{k}\,\|u\|_{1}.$ (4.11)

Let us mention that in view of the inequalities (4.18) and (4.28) which will
be proved later it follows for any smooth function $v$

$\displaystyle\|J_{1}^{\prime}v\|_{1}^{2}=\int_{\Omega}|i\nabla
J_{1}^{\prime}v+A_{\mathrm{magn}}J_{1}^{\prime}v|^{2}\,dx\,dy+\int_{\Omega}|J_{1}^{\prime}v|^{2}\,dx\,dy$
$\displaystyle\leq 2\int_{\Omega}|\nabla
J_{1}^{\prime}v|^{2}\,dxdy+2\||A_{\mathrm{magn}}|\|_{L^{\infty}(\Omega)}^{2}\int_{\Omega}|J_{1}^{\prime}v|^{2}\,dx\,dy\leq\overline{C}(\varepsilon)\|v\|_{1}^{2},$

where $\overline{C}(\varepsilon)$ is some constant.

Therefore using the density of space $H_{1}$ we are able to define

$J_{1}^{\prime}u=\underset{k\to\infty}{\mathrm{lim}}\,J_{1}^{\prime}v_{\eta_{k}}.$
(4.12)

Step (ii). The conditions (3.1)-(3.7) hold for the mappings
$J,J^{\prime},J_{1},J_{1}^{\prime}$.

Indeed, we have that the estimates (3.1)-(3.4) are satisfied with $\delta=0$.

We now prove (3.5), i.e. under the assumptions stated in Theorem 2.4
inequality (3.5) is satisfied with $\delta=\mathcal{O}(\sqrt{\varepsilon})$
for small enough $\varepsilon$.

In view of our construction we have

$\displaystyle\|f-J^{\prime}Jf\|_{0}^{2}=\int_{\Omega}|f-J^{\prime}f\chi_{\Omega_{K_{\varepsilon}}}|^{2}\,dx\,dy=\int_{\Omega}|f-f\chi_{\Omega_{K_{\varepsilon}}}|^{2}\,dx\,dy$
$\displaystyle=\int_{K_{\varepsilon}}|f|^{2}\,dx\,dy\leq\int_{B_{\varepsilon}}|f|^{2}\,dx\,dy.$
(4.13)

To proceed further we need to employ the following lemma (the proof will be
given in Appendix) applied with $\Gamma_{\varepsilon}=\emptyset$:

######  Lemma 4.1.

Suppose the assumptions of Theorem 2.1. To avoid the misunderstanding we
denote the compact hole inside of $B_{\varepsilon}$ by $\Gamma_{\varepsilon}$.
Then for any function
$u\in\mathcal{H}^{1}(\mathbb{B}_{\epsilon}\setminus\Gamma_{\varepsilon})$ the
following inequality takes place

$\int_{\mathbb{B}_{\epsilon}\setminus\Gamma_{\varepsilon}}|u(x,y)|^{2}\,dx\,dy\leq
C_{1}\varepsilon\int_{\Omega\setminus\Gamma_{\varepsilon}}(|i\nabla
u+A_{\mathrm{magn}}u|^{2}+|u|^{2})\,dx\,dy,$

where $C_{1}>0$ is a constant depending on the distance between the boundary
of $\mathbb{B}_{\epsilon}$ and the boundary of $\Omega$ and the magnetic
potential $A_{\mathrm{magn}}$.

By Lemma 4.1 and (4.13) we prove (3.5).

Using (4.12) and the completeness of $C^{\infty}(\Omega_{K_{\varepsilon}})$ in
Sobolev space $\mathcal{H}^{1}(\Omega_{K_{\varepsilon}})$ it is enough to
prove (3.6) for $u\in C^{\infty}(\Omega_{K_{\varepsilon}})$.

Taking into account that $J^{\prime}u=0$ on $K_{\varepsilon}$ and
$J_{1}^{\prime}u=u$ on $\Omega\setminus\mathbb{B}_{\epsilon}$ one has

$\displaystyle\|J^{\prime}u-J_{1}^{\prime}u\|_{0}^{2}=\int_{\Omega_{K_{\varepsilon}}}|u-J_{1}^{\prime}u|^{2}\,dx\,dy+\int_{K_{\varepsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy$
$\displaystyle=\int_{\mathbb{B}_{\epsilon}\setminus
K_{\varepsilon}}|u-J_{1}^{\prime}u|^{2}\,dx\,dy+\int_{K_{\varepsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy$
$\displaystyle\leq 2\int_{\mathbb{B}_{\epsilon}\setminus
K_{\varepsilon}}|u|^{2}\,dx\,dy+2\int_{\mathbb{B}_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy+\int_{K_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy.$
(4.14)

To estimate the first integral of the right-hand side of (4.14) we apply Lemma
4.1 with $\Gamma_{\varepsilon}=K_{\varepsilon}$.

Let us now investigate the second term in the right-hand side of (4.14).
Passing to polar coordinates one gets

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy=\int_{0}^{\epsilon}\int_{0}^{2\pi}r\left|\frac{r}{\epsilon}\tilde{u}(\epsilon,\varphi)\right|^{2}\,dr\,d\varphi$
$\displaystyle\leq\int_{0}^{\epsilon}\int_{0}^{2\pi}r|\tilde{u}(\epsilon,\varphi)|^{2}\,dr\,d\varphi\leq\epsilon\int_{0}^{\epsilon}\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}\,dr\,d\varphi$
$\displaystyle=\epsilon^{2}\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}\,d\varphi.$

Taking into account that
$\epsilon\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}\,d\varphi$ coincides
with the curvilinear integral
$\int_{\partial\mathbb{B}_{\epsilon}}|u|_{\partial\mathbb{B}_{\epsilon}}^{2}\,d\mu$,
where $|u|_{\partial\mathbb{B}_{\epsilon}}$ is the trace of $|u|$ on the
circle $\partial\mathbb{B}_{\epsilon}$, the above bound performs to

$\int_{\mathbb{B}_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy\leq\epsilon\int_{\partial\mathbb{B}_{\epsilon}}|u|_{\partial\mathbb{B}_{\epsilon}}^{2}\,d\mu.$
(4.15)

The suitable bound for the above estimate is guaranteed by the following trace
inequality [4]:

######  Lemma 4.2.

Let $v\in\mathcal{H}^{1}(\Omega\backslash\mathbb{B})$, where $\mathbb{B}$ is a
ball with center in zero and radius $\tau>0$. Then there exists a constant
$C_{2}>0$ depending on the diameter of $\Omega$ and the distance between the
boundary of $\mathbb{B}$ and the boundary of $\Omega$ such that

$\int_{\partial\mathbb{B}}|v|_{\partial\mathbb{B}}^{2}\,d\mu\leq
C_{2}\int_{\Omega\backslash\mathbb{B}}(|\nabla v|^{2}+|v|^{2})\,dx\,dy.$

Let us show that for every
$v\in\mathcal{H}^{1}(\omega),\,\omega\subset\mathbb{R}^{2}$, and a magnetic
vector potential $A$ one has

$\int_{\omega}(|\nabla v|^{2}+|v|^{2})\,dx\,dy\leq
C_{3}(A)\int_{\omega}(|i\nabla v+Av|^{2}+|v|^{2})\,dx\,dy,$ (4.16)

where $C_{3}(A)$ is given by (4.17).

Indeed, since

$\int_{\omega}|i\nabla v+Av|^{2}\,dx\,dy\geq\frac{1}{2}\int_{\omega}|\nabla
v|^{2}\,dx\,dy-2\||A|\|^{2}_{\infty}\int_{\omega}|v|^{2}\,dx\,dy$

then

$\int_{\omega}(|\nabla v|^{2}+|v|^{2})\,dx\,dy\leq 2\int_{\omega}|i\nabla
v+Av|^{2}\,dx\,dy+(4\||A|\|_{\infty}^{2}+1)\int_{\omega}|v|^{2}\,dx\,dy,$

which establishes our aim with

$C_{3}(A)=\mathrm{max}\\{2,4\||A|\|_{\infty}^{2}+1\\}.$ (4.17)

Inequality (4.15) together with Lemma 4.2, (4.16) and fact that $\epsilon\leq
2\varepsilon$ gives

$\int_{\mathbb{B}_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy\leq
2C_{2}C_{3}\varepsilon\|u\|_{1}^{2},$ (4.18)

where

$C_{3}=C_{3}(A_{\mathrm{magn}}).$ (4.19)

Since

$\int_{K_{\varepsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy\leq\int_{B_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy$

then

$\int_{K_{\varepsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy\leq
2C_{2}C_{3}\varepsilon\|u\|_{1}^{2}.$

Combining the inequality above together with (4.18) and Lemma 4.1 the right-
hand side of (4.14) can be estimated as follows

$\|J^{\prime}u-J_{1}^{\prime}u\|_{0}^{2}\leq
2C_{2}C_{3}\varepsilon\|u\|_{1}^{2}$

which proves (3.6) with $\delta=\mathcal{O}(\sqrt{\varepsilon})$.

We now give the proof of the estimate (3.7), i.e. under the assumptions stated
in Theorem 2.4 inequality (3.7) takes place with $k=2$ and
$\delta=\mathcal{O}(\varepsilon^{1/6})$ for small enough $\varepsilon$.
We have

$\displaystyle|a(f,J_{1}^{\prime}u)-a^{\prime}(J_{1}f,u)|$
$\displaystyle=\left|\int_{\Omega}(i\nabla
f+A_{{\mathrm{magn}}}f)\overline{(i\nabla(J_{1}^{\prime}u)+A_{{\mathrm{magn}}}J_{1}^{\prime}u)}\,dx\,dy-\int_{\Omega_{K_{\varepsilon}}}(i\nabla(J_{1}f)+A_{{\mathrm{magn}}}J_{1}f)\overline{(i\nabla
u+A_{{\mathrm{magn}}}u)}\,dx\,dy\right|$
$\displaystyle=\biggl{|}\int_{\Omega_{\mathbb{B}_{\epsilon}}}(i\nabla
f+A_{{\mathrm{magn}}}f)\overline{(i\nabla
u+A_{{\mathrm{magn}}}u)}\,dx\,dy+\int_{\mathbb{B}_{\epsilon}}(i\nabla
f+A_{{\mathrm{magn}}}f)\overline{(i\nabla(J_{1}^{\prime}u)+A_{{\mathrm{magn}}}J_{1}^{\prime}u)}\,dx\,dy$
$\displaystyle-\int_{\Omega_{\mathbb{B}_{\epsilon}}}(i\nabla
f+A_{{\mathrm{magn}}}f)\overline{(i\nabla
u+A_{{\mathrm{magn}}}u)}\,dx\,dy-\int_{\mathbb{B}_{\epsilon}\backslash
K_{\varepsilon}}(i\nabla f+A_{{\mathrm{magn}}}f)\overline{(i\nabla
u+A_{{\mathrm{magn}}}u)}\,dx\,dy\biggr{|}$
$\displaystyle=\biggl{|}\int_{\mathbb{B}_{\epsilon}}(i\nabla
f+A_{{\mathrm{magn}}}f)\overline{(i\nabla(J_{1}^{\prime}u)+A_{{\mathrm{magn}}}J_{1}^{\prime}u)}\,dx\,dy-\int_{\mathbb{B}_{\epsilon}\backslash
K_{\varepsilon}}(i\nabla f+A_{{\mathrm{magn}}}f)\overline{(i\nabla
u+A_{{\mathrm{magn}}}u)}\,dx\,dy\biggr{|}$
$\displaystyle\leq\left|\int_{\mathbb{B}_{\epsilon}}(i\nabla
f+A_{{\mathrm{magn}}}f)\overline{(i\nabla(J_{1}^{\prime}u)+A_{{\mathrm{magn}}}J_{1}^{\prime}u)}\,dx\,dy\right|+\left|\int_{\mathbb{B}_{\epsilon}\backslash
K_{\varepsilon}}(i\nabla f+A_{{\mathrm{magn}}}f)\overline{(i\nabla
u+A_{{\mathrm{magn}}}u)}\,dx\,dy\right|.$ (4.20)

Let us estimate each term of (4.20). Starting from the first term one gets

$\displaystyle\left|\int_{\mathbb{B}_{\epsilon}}(i\nabla
f+A_{{\mathrm{magn}}}f)\overline{(i\nabla(J_{1}^{\prime}u)+A_{{\mathrm{magn}}}J_{1}^{\prime}u)}\,dx\,dy\right|$
$\displaystyle\leq\left(\int_{\mathbb{B}_{\epsilon}}|i\nabla
f+A_{{\mathrm{magn}}}f|^{2}\,dx\,dy\right)^{1/2}\left(\int_{\mathbb{B}_{\epsilon}}|i\nabla(J_{1}^{\prime}u)+A_{{\mathrm{magn}}}J_{1}^{\prime}u|^{2}\,dx\,dy\right)^{1/2}.$
(4.21)

Dealing with the first term of the right-hand side of the above inequality we
have

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|i\nabla
f+A_{{\mathrm{magn}}}f|^{2}\,dx\,dy\leq 2\int_{\mathbb{B}_{\epsilon}}|\nabla
f|^{2}\,dx\,dy+2\int_{\mathbb{B}_{\epsilon}}|A_{{\mathrm{magn}}}|^{2}|f|^{2}\,dx\,dy$
$\displaystyle\leq 2\int_{B_{\epsilon}}|\nabla
f|^{2}\,dx\,dy+2\||A_{{\mathrm{magn}}|}\|^{2}_{\infty}\int_{B_{\epsilon}}|f|^{2}\,dx\,dy.$
(4.22)

The first and second terms in (4.22) can be estimated via the following lemmas
(the proofs are given in Appendix):

######  Lemma 4.3.

For any function $g\in\mathrm{Dom}(H_{\Omega}(A_{\mathrm{magn}}))$ the
following estimate takes place

$\int_{\mathbb{B}_{\epsilon}}|\nabla g|^{2}\,dx\,dy\leq
C_{5}\epsilon^{4/3}\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}g+g|^{2}\,dx\,dy,$

with the constant $C_{5}$ depends on $\Omega$ and magnetic potential
$A_{\mathrm{magn}}$.

######  Lemma 4.4.

For any $g\in\mathrm{Dom}(H_{\Omega}(A_{\mathrm{magn}}))$ the following
inequality takes place

$\int_{\mathbb{B}_{\epsilon}}|g|^{2}\,dx\,dy\leq
C_{6}\epsilon^{4/3}\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}g+g|^{2}\,dx\,dy,$

with the constant $C_{6}$ depends on $\Omega$ and magnetic potential
$A_{\mathrm{magn}}$.

This together with the fact that $\epsilon\leq 2\varepsilon$ estimates the
right-hand side of (4.22) as follows

$\int_{\mathbb{B}_{\epsilon}}|i\nabla f+A_{{\mathrm{magn}}}f|^{2}\,dx\,dy\leq
2^{4/3}C_{7}\varepsilon^{4/3}\int_{\Omega}|(i\nabla+A_{{\mathrm{magn}}})^{2}f+f|^{2}\,dx\,dy=2^{4/3}C_{7}\varepsilon^{4/3}\|f\|_{2}^{2}$
(4.23)

with $C_{7}=2C_{5}+2\||A_{{\mathrm{magn}}}|\|_{\infty}^{2}C_{6}$.

To proceed further with the proof of an upper bound of (4.21) we need to
estimate integral
$\int_{\mathbb{B}_{\epsilon}}|i\nabla(J_{1}^{\prime}u)+A_{{\mathrm{magn}}}J_{1}^{\prime}u|^{2}\,dx\,dy$.
One has

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|i\nabla(J_{1}^{\prime}u)+A_{{\mathrm{magn}}}J_{1}^{\prime}u|^{2}\,dx\,dy\leq
2\int_{\mathbb{B}_{\epsilon}}|\nabla(J_{1}^{\prime}u)|^{2}\,dx\,dy+2\int_{\mathbb{B}_{\epsilon}}|A_{{\mathrm{magn}}}|^{2}|J_{1}^{\prime}u|^{2}\,dx\,dy$
$\displaystyle\leq
2\int_{\mathbb{B}_{\epsilon}}|\nabla(J_{1}^{\prime}u)|^{2}\,dx\,dy+2\||A_{{\mathrm{magn}}}|\|_{\infty}^{2}\int_{\mathbb{B}_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy.$
(4.24)

Passing to polar coordinates in the first integral in the right-hand side of
(4.24) one gets

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|\nabla
J_{1}^{\prime}u|^{2}\,dx\,dy=\int_{\mathbb{B}_{\epsilon}}\left(\left|\frac{\partial(J_{1}^{\prime}u)}{\partial
x}\right|^{2}+\left|\frac{\partial(J_{1}^{\prime}u)}{\partial
y}\right|^{2}\right)\,dx\,dy$
$\displaystyle=\int_{0}^{\epsilon}\int_{0}^{2\pi}r\left|\frac{1}{\epsilon}\tilde{u}(\epsilon,\varphi)\,\cos\varphi-\frac{1}{\epsilon}\frac{\partial\tilde{u}}{\partial\varphi}(\epsilon,\varphi)\,\sin\varphi\right|^{2}\,dr\,d\varphi$
$\displaystyle+\int_{0}^{\varepsilon}\int_{0}^{2\pi}r\left|\frac{1}{\epsilon}\tilde{u}(\epsilon,\varphi)\,\sin\varphi+\frac{1}{\epsilon}\frac{\partial\tilde{u}}{\partial\varphi}(\epsilon,\varphi)\,\cos\varphi\right|^{2}\,dr\,d\varphi$
$\displaystyle\leq\frac{4}{\epsilon}\int_{0}^{\epsilon}\int_{0}^{2\pi}\left(|\tilde{u}(\epsilon,\varphi)|^{2}+\left|\frac{\partial\tilde{u}}{\partial\varphi}(\epsilon,\varphi)\right|^{2}\right)\,dr\,d\varphi$
$\displaystyle=4\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}d\varphi+4\int_{0}^{2\pi}\left|\frac{\partial\tilde{u}}{\partial\varphi}(\epsilon,\varphi)\right|^{2}\,d\varphi.$
(4.25)

As in proof of (3.6) we note that

$4\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}\,d\varphi=\frac{4}{\epsilon}\,\int_{\partial\mathbb{B}_{\epsilon}}|u|_{\partial\mathbb{B}_{\epsilon}}^{2}\,d\mu,$

which can be estimated using Lemma 4.2 and (4.16) as follows:

$4\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}\,d\varphi\leq\frac{4}{\epsilon}C_{4}C_{3}\|u\|_{1}^{2}.$
(4.26)

To obtain the upper bound for the second term of 4.25 we use the following
auxiliary lemma (the proof is given in Appendix):

######  Lemma 4.5.

Let $\mathbb{B}_{2\varepsilon}$ and $\mathbb{B}_{\varepsilon}$ be the balls
with center in origin and radii $\varepsilon$ and $2\varepsilon$. Let
$g\in\mathcal{H}^{1}(\mathbb{B}_{2\varepsilon}\setminus\mathbb{B}_{\varepsilon})$.
Then there exists $\tau\in(\varepsilon,2\varepsilon)$ such that

$\int_{0}^{2\pi}\left|\frac{\partial\tilde{g}}{\partial\varphi}(\tau,\varphi)\right|^{2}\,d\varphi\leq
4\int_{\mathbb{B}_{2\varepsilon}\setminus\mathbb{B}_{\varepsilon}}(|\nabla
g|^{2}+|g|^{2})\,dx\,dy,$

where $\tilde{g}(r,\varphi):=g(r\cos\varphi,r\sin\varphi)$.

Now we are able to choose the number $\epsilon$. Let us apply Lemma 4.5 for
$g=u$. In case if mentioned in above lemma $\tau$ belongs to interval
$(\varepsilon,3\varepsilon/2]$ then we choose $\epsilon$ as the supremum of
all such numbers in $(\varepsilon,3\varepsilon/2]$. In the opposite case if
$\tau\in(3\varepsilon/2,2\varepsilon)$ then let $\epsilon$ be the infimum of
such numbers. Since $u$ is smooth function then inequality (4.5) is satisfied
with $\tau=\epsilon$.

Combining this together with (4.16) one gets

$\int_{0}^{2\pi}\left|\frac{\partial\tilde{u}}{\partial\varphi}(\epsilon,\varphi)\right|^{2}\,d\varphi\leq
4C_{3}\|u\|_{1}^{2}.$ (4.27)

Hence, by virtue of (4.25), (4.26) and (4.27)

$\int_{\mathbb{B}_{\epsilon}}|\nabla
J_{1}^{\prime}u|^{2}\,dx\,dy\leq\frac{4}{\epsilon}(C_{4}C_{3}+16C_{3})\|u\|_{1}^{2}.$
(4.28)

In view of (4.18), (4.24), (4.28) and the fact that
$\epsilon\geq\varepsilon/2$ we arrive

$\int_{\mathbb{B}_{\epsilon}}|i\nabla(J_{1}^{\prime}u)+A_{{\mathrm{magn}}}J_{1}^{\prime}u|^{2}\,dx\,dy\leq\left(\frac{8}{\epsilon}(C_{4}C_{3}+16C_{3})+4\||A_{{\mathrm{magn}}}|\|_{\infty}^{2}C_{4}C_{3}\varepsilon\right)\|u\|_{1}^{2}.$
(4.29)

Finally, employing (4.23) and the above estimate in inequality (4.21) it
follows that

$\left|\int_{\mathbb{B}_{\epsilon}}(i\nabla
f+A_{{\mathrm{magn}}}f)\overline{(i\nabla(J_{1}^{\prime}u)+A_{{\mathrm{magn}}}J_{1}^{\prime}u)}\,dx\,dy\right|\\\
\leq 2^{13/6}(C_{7}C_{3}(C_{4}+16))^{1/2}\varepsilon^{1/6}\|f\|_{2}\|u\|_{1}.$

Let us now consider the second term in (4.20). By virtue of (4.23) one has

$\displaystyle\left|\int_{\mathbb{B}_{\epsilon}\backslash
K_{\varepsilon}}(i\nabla f+A_{{\mathrm{magn}}}f)\overline{(i\nabla
u+A_{{\mathrm{magn}}}u)}\,dx\,dy\right|$
$\displaystyle\leq\left(\int_{\mathbb{B}_{\epsilon}}|i\nabla
f+A_{{\mathrm{magn}}}f|^{2}dxdx\right)^{1/2}\left(\int_{\mathbb{B}_{\epsilon}\backslash
K_{\varepsilon}}|i\nabla u+A_{{\mathrm{magn}}}u|^{2}\,dx\,dy\right)^{1/2}$
$\displaystyle\leq 2^{2/3}(C_{5})^{1/2}\varepsilon^{2/3}\|f\|_{2}\|u\|_{1}.$

Hence the right-hand side of inequality (3.7) for small enough $\varepsilon$
satisfies

$\mathrm{r.h.s.}(\ref{7})=\mathcal{O}\left(\varepsilon^{1/6}\right)\,\|f\|_{2}\,\|u\|_{1},$

which ends the proof.∎

##  5 Appendix

In this section we give the proofs of Lemmas 4.1, 4.3-4.5. Since the outline
of our proofs are a slight modification of the proofs in [1, Appendix],
therefore we refer [1] for identical parts.

###  5.1 Proof of Lemma 4.1

We are going to use the following result:

######  Lemma 5.1.

[The proof will be given in the end of this section] Let
$\Omega_{\Gamma_{\varepsilon}}$ be an open domain satisfying property∗. Then
for any $u\in\mathcal{H}^{1}(\Omega_{\Gamma_{\varepsilon}})$ and for almost
any $(x_{0},y_{0})\in\Omega_{\Gamma_{\varepsilon}}$ the following statement
takes place:

$\displaystyle|u(x_{0},y_{0})|\leq
C_{8}\left(\int_{l(x_{0})\setminus\Gamma_{\varepsilon}}\left|\frac{\partial
u}{\partial
y}(x_{0},z)\right|^{2}\,dz+\int_{l(x_{0})\setminus\Gamma_{\varepsilon}}|u(x_{0},z)|^{2}\,dz\right)^{1/2},$
(5.1) $\displaystyle\text{if}\quad(x_{0},y_{0})\quad\text{satisfies first
condition of {property${}^{*}$}}$ $\displaystyle|u(x_{0},y_{0})|\leq
C_{8}\left(\int_{h(y_{0})\setminus\Gamma_{\varepsilon}}\left|\frac{\partial
u}{\partial
z}(z,y_{0})\right|^{2}+\int_{h(y_{0})\setminus\Gamma_{\varepsilon}}|u(z,y_{0})|^{2}\,dz\right)^{1/2},$
(5.2) $\displaystyle\text{if}\quad(x_{0},y_{0})\quad\text{satisfies second
condition of {property${}^{*}$}},$

where $C_{8}>0$ is a constant depending on the diameter of $\Omega$ and the
distance between point $(x_{0},y_{0})$ and the boundary of $\Omega$.

In view of the mentioned result for any
$(x,y)\in\mathbb{B}_{\epsilon}\backslash\Gamma_{\varepsilon}$

$\displaystyle|u(x,y)|^{2}\leq
C_{8}^{2}\left(\int_{l(x)\setminus\Gamma_{\varepsilon}}\left|\frac{\partial
u}{\partial
y}(x,z)\right|^{2}\,dz+\int_{l(x)\setminus\Gamma_{\varepsilon}}|u(x,z)|^{2}\,dz\right)$
$\displaystyle+C_{8}^{2}\left(\int_{h(y)\setminus\Gamma_{\varepsilon}}\left|\frac{\partial
u}{\partial
z}(z,y)\right|^{2}\,dz+\int_{h(y)\setminus\Gamma_{\varepsilon}}|u(z,y)|^{2}\,dz\right).$

Therefore we obtain

$\displaystyle\int_{\mathbb{B}_{\epsilon}\backslash\Gamma_{\varepsilon}}|u(x,y)|^{2}\,dx\,dy\leq
C_{8}^{2}\int_{\mathbb{B}_{\epsilon}\backslash\Gamma_{\varepsilon}}\int_{l(x)\setminus\Gamma_{\varepsilon}}\left(\left|\frac{\partial
u}{\partial z}(x,z)\right|^{2}+|u(x,z)|^{2}\right)\,dz\,dx\,dy$
$\displaystyle+C_{8}^{2}\int_{\mathbb{B}_{\epsilon}\backslash\Gamma_{\varepsilon}}\int_{h(y)\setminus\Gamma_{\varepsilon}}\left(\left|\frac{\partial
u}{\partial z}(z,y)\right|^{2}+|u(z,y)|^{2}\right)\,dz\,dx\,dy$
$\displaystyle\leq
C_{8}^{2}\int_{-\epsilon}^{\epsilon}\int_{\\{y:(x,y)\in\mathbb{B}_{\epsilon}\backslash\Gamma_{\varepsilon}\\}}\int_{l(x)\setminus\Gamma_{\varepsilon}}\left(\left|\frac{\partial
u}{\partial z}(x,z)\right|^{2}+|u(x,z)|^{2}\right)\,dz\,dx\,dy$
$\displaystyle+C_{8}^{2}\int_{-\epsilon}^{\epsilon}\int_{\\{x:(x,y)\in\mathbb{B}_{\epsilon}\backslash\Gamma_{\varepsilon}\\}}\int_{h(y)\setminus\Gamma_{\varepsilon}}\left(\left|\frac{\partial
u}{\partial z}(z,y)\right|^{2}+|u(z,y)|^{2}\right)\,dz\,dy\,dx$
$\displaystyle\leq
2C_{8}^{2}\epsilon\int_{\Omega\backslash\Gamma_{\varepsilon}}(|\nabla
u|^{2}+2|u|^{2})\,dx\,dy\leq
4C_{8}^{2}\epsilon\int_{\Omega\backslash\Gamma_{\varepsilon}}(|\nabla
u|^{2}+|u|^{2})\,dx\,dy.$

This together with (4.16) and the fact that $\epsilon\leq 2\varepsilon$ proves
the lemma.

∎

###  5.2 Proof of Lemma 4.3

To proceed with a proof we need the following auxiliary material [6, Lemma
4.9]:

######  Lemma 5.2.

Let $\Pi\subset\mathbb{R}^{n}$ be a convex set and let $G$ and $Q$ be
arbitrary measurable sets in $\Pi$ with $\mu\,(G)\neq 0$. Then, for all
$v\in\mathcal{H}^{1}(\Pi)$, the following inequality holds:

$\displaystyle\int_{Q}|v|^{2}\,dx\,dy$ (5.1)
$\displaystyle\leq\frac{2\mu\,(Q)}{\mu\,(G)}\int_{G}|v|^{2}\,dx\,dy+\frac{C(n)(d(\Pi))^{n+1}(\mu\,(Q))^{1/n}}{\mu\,(G)}\int_{\Pi}|\nabla
v|^{2}\,dx\,dy,$

where $d(\Pi)$ is the parameter of $\Pi$, $\mu$ is the Lebesque measure on
$\mathbb{R}^{n}$, and the constant $C(n)$ depends only on the dimension of
$\mathbb{R}^{n}$.

Let $\Pi\subset\mathbb{R}^{n}$ be a convex set and let $G$ and $Q$ be
arbitrary measurable sets in $\Pi$ with $\mu\,(G)\neq 0$. Applying Lemma(5.2)
for $Q=\mathbb{B}_{\epsilon}$ and $G=\Pi=G_{\epsilon}$ we have

$\displaystyle\int_{\mathbb{B}_{\epsilon}}\left|\frac{\partial g}{\partial
x}\right|^{2}\,dx\,dy\leq\frac{2\mu\,(\mathbb{B}_{\epsilon})}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\frac{\partial
g}{\partial x}\right|^{2}\,dx\,dy$
$\displaystyle+\frac{C(2)(d\,(G_{\epsilon}))^{3}(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\nabla\left(\frac{\partial
g}{\partial x}\right)\right|^{2}\,dx\,dy$
$\displaystyle=\frac{2\mu\,(\mathbb{B}_{\epsilon})}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\frac{\partial
g}{\partial
x}\right|^{2}\,dx\,dy+\frac{C(2)(d\,(G_{\epsilon}))^{3}(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left(\left|\frac{\partial^{2}g}{\partial
x^{2}}\right|^{2}+\left|\frac{\partial^{2}g}{\partial x\partial
y}\right|^{2}\right)\,dx\,dy$

and

$\displaystyle\int_{\mathbb{B}_{\epsilon}}\left|\frac{\partial g}{\partial
y}\right|^{2}\,dx\,dy\leq\frac{2\mu\,(\mathbb{B}_{\epsilon})}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\frac{\partial
g}{\partial y}\right|^{2}\,dx\,dy$
$\displaystyle+\frac{C(2)(d\,(G_{\epsilon}))^{3}(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\nabla\left(\frac{\partial
g}{\partial y}\right)\right|^{2}\,dx\,dy$
$\displaystyle=\frac{2\mu\,(\mathbb{B}_{\epsilon})}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\frac{\partial
g}{\partial
y}\right|^{2}\,dx\,dy+\frac{C(2)(d\,(G_{\epsilon}))^{3}(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left(\left|\frac{\partial^{2}g}{\partial
y\partial x}\right|^{2}+\left|\frac{\partial^{2}g}{\partial
y^{2}}\right|^{2}\right)\,dx\,dy.$

Combining the above inequalities we arrive

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|\nabla
g|^{2}\,dx\,dy\leq\frac{2\mu\,(\mathbb{B}_{\epsilon})}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}|\nabla
g|^{2}\,dx\,dy$
$\displaystyle+\frac{C(2)(d\,(G_{\epsilon}))^{3}(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left(\left|\frac{\partial^{2}g}{\partial
x^{2}}\right|^{2}+2\left|\frac{\partial^{2}g}{\partial x\partial
y}\right|^{2}+\left|\frac{\partial^{2}f}{\partial
y^{2}}\right|^{2}\right)\,dx\,dy$
$\displaystyle\leq\frac{(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\left(2(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}+C(2)(d\,(G_{\epsilon}))^{3}\right)\|g\|_{\mathcal{H}^{2}(G_{\epsilon})}^{2}$
$\displaystyle=\frac{\sqrt{\pi}\epsilon}{\mu\,(G_{\epsilon})}\left(2\sqrt{\pi}\epsilon+C(2)(d\,(G_{\epsilon}))^{3}\right)\|g\|_{\mathcal{H}^{2}(G_{\epsilon})}^{2}.$

Let us choose $G_{\epsilon}=\\{r:\,0\leq r\leq\epsilon^{1-\alpha}\\}$ with
some $\alpha\in\left(\frac{1}{2},1\right)$ to be chosen later. One can easily
notice that $\mathbb{B}_{\epsilon}\subset G_{\epsilon}$. The the above
inequality becomes

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|\nabla g|^{2}\,dx\,dy\leq
2\epsilon\left(\epsilon^{2\alpha-1}+\frac{4C(2)}{\sqrt{\pi}}\epsilon^{1-\alpha}\right)\,\|g\|^{2}_{\mathcal{H}^{2}(G_{\epsilon})}$
$\displaystyle\leq
2\mathrm{max}\left\\{1,\frac{4C(2)}{\sqrt{\pi}}\right\\}\epsilon\left(\epsilon^{2\alpha-1}+\epsilon^{1-\alpha}\right)\,\|g\|^{2}_{\mathcal{H}^{2}(G_{\epsilon})}.$
(5.2)

Let us investigate the function
$F(\alpha):=\epsilon^{2\alpha}+\epsilon^{2-\alpha}$ on interval
$\left(\frac{1}{2},1\right)$. It attains its minimum at
$\alpha_{0}=\frac{2}{3}-\frac{1}{3}\frac{\ln 2}{\ln\epsilon}$ and takes the
value $F(\alpha_{0})=\epsilon^{4/3}\left(\frac{1}{4^{1/3}}+2^{1/3}\right)$.

Let us now choose $\alpha=\alpha_{0}$ in inequality (5.2). Then

$\int_{\mathbb{B}_{\epsilon}}|\nabla g|^{2}\,dx\,dy\leq
2\epsilon^{4/3}\left(\frac{1}{4^{1/3}}+2^{1/3}\right)\mathrm{max}\left\\{1,\frac{4C(2)}{\sqrt{\pi}}\right\\}\,\|g\|_{\mathcal{H}^{2}(G_{\epsilon})}^{2}.$
(5.3)

We employ the interior regularity theorem [2] :

######  Theorem 5.3.

(Interior Regularity Theorem.) Suppose that $h\in\mathcal{H}^{1}(\Omega)$ is a
weak solution to $Lh=w$ where $L$ is the elliptic operator on $\Omega$ with
$C^{1}$ smooth coefficients. Suppose that $w\in L^{2}(\Omega)$. Then
$h\in\mathcal{H}^{2}_{\mathrm{loc}}(\Omega)$ and for each
$\Omega_{0}\subset\Omega$ there is a constant $c=c(\Omega_{0})$ independent of
$h$ and $w$ such that:

$\|h\|_{\mathcal{H}^{2}(\Omega_{0})}\leq
c\left(\|h\|_{L^{2}(\Omega)}+\|w\|_{L^{2}(\Omega)}\right).$ (5.4)

This together with the fact that the magnetic potential $A_{\mathrm{magn}}$ is
twice continuously differentiable on $G_{\epsilon}$ and having in mind that
$\epsilon\leq 2\varepsilon$ estimates the right-hand side of (5.2) via
$\|g\|_{2}$ as follows

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|\nabla g|^{2}\,dx\,dy$
$\displaystyle\leq
2^{10/3}\varepsilon^{4/3}c^{2}\left(\frac{1}{4^{1/3}}+2^{1/3}\right)\mathrm{max}\left\\{1,\frac{4C(2)}{\sqrt{\pi}}\right\\}\,\int_{\Omega}\left(|(i\nabla+A_{\mathrm{magn}})^{2}g|^{2}+|g|^{2}\right)\,dx\,dy.$

The above inequality together with the following lemma completes the proof:

######  Lemma 5.4.

[The proof will be given in the end of this section] For any
$z\in\mathrm{Dom}(H_{\Omega}(A_{\mathrm{magn}}))$ the following estimate is
valid

$\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}z+z|^{2}\,dx\,dy\geq\int_{\Omega}(|(i\nabla+A_{\mathrm{magn}})^{2}z|^{2}+|z|^{2})\,dx\,dy.$

∎

###  5.3 Proof of Lemma 4.4

Using the same notations as in previous subsection, Lemma 5.2 and repeating
the similar calculations one infers that

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|g|^{2}\,dx\,dy\leq
2\epsilon^{4/3}\left(\frac{1}{4^{1/3}}+2^{1/3}\right)\mathrm{max}\left\\{1,\frac{4C(2)}{\sqrt{\pi}}\right\\}\,\|g\|_{\mathcal{H}^{1}(G_{\epsilon})}^{2}$
$\displaystyle\leq
2\epsilon^{4/3}\left(\frac{1}{4^{1/3}}+2^{1/3}\right)\mathrm{max}\left\\{1,\frac{4C(2)}{\sqrt{\pi}}\right\\}\,\|g\|_{\mathcal{H}^{2}(G_{\epsilon})}^{2}.$
(5.1)

We finish the proof similarly as in previous subsection.

∎

###  5.4 Proof of Lemma 4.5

We first prove the following auxiliary statement: there exists
$\tau\in(\varepsilon,2\varepsilon)$ such that

$\int_{0}^{2\pi}\left|\nabla
g(\tau\cos\varphi,\tau\sin\varphi)\right|^{2}\,d\varphi\leq\frac{1}{\varepsilon^{2}}\|g\|_{1}^{2}.$
(5.2)

Let us assume the opposite: for any $r\in(\varepsilon,2\varepsilon)$ we have

$\int_{0}^{2\pi}\left|\nabla
g(r\cos\varphi,r\sin\varphi)\right|^{2}\,d\varphi>\frac{1}{\varepsilon^{2}}\|g\|_{1}^{2}.$

Passing to polar coordinates in integral
$\int_{\mathbb{B}_{2\varepsilon}\backslash\mathbb{B}_{\varepsilon}}|\nabla
g|^{2}\,dx\,dy$ and using the above bound we get

$\int_{\mathbb{B}_{2\varepsilon}\backslash\mathbb{B}_{\varepsilon}}|\nabla
g|^{2}\,dx\,dy=\int_{\varepsilon}^{2\varepsilon}\,\int_{0}^{2\pi}r\left|\nabla
g(r\cos\varphi,r\sin\varphi)\right|^{2}\,d\varphi\,dr>\|g\|_{1}^{2}.$

This contradicts with the fact that the left-hand side of the above inequality
does not exceed $\|g\|_{1}^{2}$. Hence there exists at least one number
$\tau\in(\varepsilon,2\varepsilon)$ such that

$\int_{0}^{2\pi}|\nabla
g(\tau\cos\varphi,\tau\sin\varphi)|^{2}\,d\varphi\leq\frac{1}{\varepsilon^{2}}\|g\|_{1}^{2}.$
(5.3)

In view of the representation of the derivative of function
$\tilde{g}(r,\varphi)=g(r\cos\varphi,r\sin\varphi)$

$\frac{\partial\widetilde{g}}{\partial\varphi}(\tau,\varphi)=-\tau\frac{\partial
g}{\partial x}(\tau\cos\varphi,\tau\sin\varphi)\sin\varphi+\tau\frac{\partial
g}{\partial y}(\tau\cos\varphi,\tau\sin\varphi)\cos\varphi$

and the Cauchy inequality we conclude that

$\left|\frac{\partial\widetilde{g}}{\partial\varphi}(\tau,\varphi)\right|\leq
2\varepsilon\left|\nabla g(\tau\cos\varphi,\tau\sin\varphi)\right|.$

Hence, employing (5.3) we establish that

$\int_{0}^{2\pi}\left|\frac{\partial\tilde{g}}{\partial\varphi}(\tau,\varphi)\right|^{2}\,d\varphi\leq
4\varepsilon^{2}\int_{0}^{2\pi}\left|\nabla
g(\tau\cos\varphi,\tau\sin\varphi)\right|^{2}\,d\varphi\leq 4\|g\|_{1}^{2},$

which concludes the proof.∎

###  5.5 Proof of Lemma 5.1

Let $(x_{0},y_{0})\in\Omega_{\Gamma_{\varepsilon}}$. Assume the validity of
the first condition of property∗. Let $y_{1}(x_{0})$ be a point of
intersection of $l(x_{0})$ and the boundary of $\Omega$. Without less of
generality suppose that $y_{1}(x_{0})<y_{0}$ and the interval
$(y_{1}(x_{0}),y_{0})$ does not contain points from $\Gamma_{\varepsilon}$.
One can easily check that there exists $y_{2}(x_{0})\in(y_{1}(x_{0}),y_{0})$
such that

$|u(x_{0},y_{0})|\leq\frac{1}{\sqrt{y_{0}-y_{1}(x_{0})}}\sqrt{\int_{y_{1}(x_{0})}^{y_{0}}|u(x_{0},z)|^{2}\,dz}.$

Therefore

$\displaystyle|u(x_{0},y_{0})|^{2}=\left|u(x_{0},y_{2}(x_{0}))+\int_{y_{2}(x_{0})}^{y_{0}}\frac{\partial
u}{\partial z}(x_{0},z)\,dz\right|^{2}$ $\displaystyle\leq
2|u(x_{0},y_{2}(x_{0}))|^{2}+2(y_{0}-y_{2}(x_{0}))\int_{y_{2}(x)}^{y}\left|\frac{\partial
u}{\partial z}(x,z)\right|^{2}\,dy$
$\displaystyle\leq\frac{2}{y_{0}-y_{1}(x_{0})}\int_{y_{1}(x_{0})}^{y_{0}}|u(x_{0},z)|^{2}\,dz+2(y_{0}-y_{2}(x_{0}))\int_{y_{2}(x_{0})}^{y_{0}}\left|\frac{\partial
u}{\partial z}(x_{0},z)\right|^{2}\,dz$
$\displaystyle\leq\frac{2}{\mathrm{dist}((x_{0},y_{0}),\partial\Omega)}\int_{y_{1}(x_{0})}^{y_{0}}|u(x_{0},z)|^{2}\,dz+2\mathrm{diam}(\Omega)\int_{y_{1}(x_{0})}^{y_{0}}\left|\frac{\partial
u}{\partial z}(x_{0},z)\right|^{2}\,dz$
$\displaystyle\leq\frac{2}{\mathrm{dist}((x_{0},y_{0}),\partial\Omega)}\int_{l(x_{0})\setminus\Gamma_{\varepsilon}}|u(x_{0},z)|^{2}\,dz+2\mathrm{diam}(\Omega)\int_{l(x_{0})\setminus\Gamma_{\varepsilon}}\left|\frac{\partial
u}{\partial z}(x_{0},z)\right|^{2}\,dz,$

where $\mathrm{diam}(\Omega)$ is the diameter of $\Omega$ and
$\mathrm{dist}((x_{0},y_{0}),\partial\Omega)$ is the distance between
$(x_{0},y_{0})$ and the boundary of $\Omega$. This proves (5.1) with

$C_{8}=\left(\mathrm{max}\left\\{\frac{2}{\mathrm{dist}((x_{0},y_{0}),\partial\Omega)},\,2\mathrm{diam}(\Omega)\right\\}\right)^{/2}.$

The case when $(x_{0},y_{0})$ satisfies second condition of property∗ can be
studied similarly. Repeating the same ideas one gets the validity of (5.2).∎

###  5.6 Proof of Lemma 5.4

It is straightforward to check that

$\displaystyle\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u+u|^{2}\,dx\,dy$
$\displaystyle=\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u|^{2}\,dx\,dy+2\int_{\Omega}|i\nabla
u+A_{\mathrm{magn}}u|^{2}\,dx\,dy+\int_{\Omega}|u|^{2}\,dx\,dy$
$\displaystyle\geq\int_{\Omega}|u|^{2}\,dx\,dy.$ (5.4)

Let us consider two cases:

$\displaystyle\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u|^{2}\,dx\,dy\geq
4\int_{\Omega}|u|^{2}\,dx\,dy,$ (5.5)
$\displaystyle\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u|^{2}\,dx\,dy<4\int_{\Omega}|u|^{2}\,dx\,dy.$
(5.6)

Starting from the first one and employing (5.4) we have

$\displaystyle\sqrt{\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u+u|^{2}\,dx\,dy}\geq\sqrt{\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u|^{2}\,dx\,dy}-\sqrt{\int_{\Omega}|u|^{2}\,dx\,dy}$
$\displaystyle\geq\frac{1}{2}\sqrt{\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u|^{2}\,dx\,dy}\geq\frac{1}{4}\sqrt{\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u|^{2}\,dx\,dy}+\frac{1}{2}\sqrt{\int_{\Omega}|u|^{2}\,dx\,dy}$
$\displaystyle\geq\frac{1}{4}\left(\sqrt{\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u|^{2}\,dx\,dy}+\sqrt{\int_{\Omega}|u|^{2}\,dx\,dy}\right).$

Hence we arrive at the bound

$\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u+u|^{2}\,dx\,dy\geq\frac{1}{16}\left(\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u|^{2}\,dx\,dy+\int_{\Omega}|u|^{2}\,dx\,dy\right).$
(5.7)

Now let us consider the case (5.6). In view of inequality (5.4) we conclude

$\displaystyle\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u+u|^{2}\,dx\,dy\geq\int_{\Omega}|u|^{2}\,dx\,dy\geq\frac{1}{2}\int_{\Omega}|u|^{2}\,dx\,dy+\frac{1}{8}\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u|^{2}\,dx\,dy$
$\displaystyle\geq\frac{1}{8}\left(\int_{\Omega}|(i\nabla+A_{\mathrm{magn}})^{2}u|^{2}\,dx\,dy+\int_{\Omega}|u|^{2}\,dx\,dy\right).$

Combining the above estimate together with (5.7) we complete the proof of the
lemma. ∎

## Funding

The research of D.B. and B.S. was supported by the Czech-Polish project
BPI/PST/2021/1/00031. Furthermore, D.B. wants to thank for the support to the
Czech Science Foundation (GACR) within the project 21-07129S.

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